Discontinuity, Nonlinearity, and Complexity

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Mar 1, 2017 - Diagonal 647 (ETSEIB,UPC). Email: [email protected] Marian Gidea. Department of Mathematical Sciences. Yeshiva University.

Volume 6 Issue 1 March 2017

ISSN 2164‐6376 (print) ISSN 2164‐6414 (online) 

An Interdisciplinary Journal of

Discontinuity, Nonlinearity, and Complexity

Discontinuity, Nonlinearity, and Complexity Editors Valentin Afraimovich San Luis Potosi University, IICO-UASLP, Av.Karakorum 1470 Lomas 4a Seccion, San Luis Potosi, SLP 78210, Mexico Fax: +52 444 825 0198 Email: [email protected]

Lev Ostrovsky University of Colorado, Boulder, and University of North Carolina, Chapel Hill, USA Email: [email protected]

Xavier Leoncini Centre de Physique Théorique, Aix-Marseille Université, CPT Campus de Luminy, Case 907 13288 Marseille Cedex 9, France Fax: +33 4 91 26 95 53 Email: [email protected]

Dimitri Volchenkov Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA & Sichuan University of Science and Engineering, Sichuan, Zigong 643000, China Email: [email protected]

Associate Editors Marat Akhmet Department of Mathematics Middle East Technical University 06531 Ankara, Turkey Fax: +90 312 210 2972 Email: [email protected]

Ranis N. Ibragimov Department of Mathematics and Physics University of Wisconsin-Parkside 900 Wood Rd, Kenosha, WI 53144 Tel: 1(262) 595-2517 Email: [email protected]

J. A. Tenreiro Machado Institute of Engineering, Polytechnic of Porto, Dept. of Electrical Engineering, Rua Dr. Antonio Bernardino de Almeida, 431, 4249-015 Porto, Portugal Fax: 351-22-8321159 Email: [email protected]

Dumitru Baleanu Department of Mathematics Cankaya University, Balgat 06530 Ankara, Turkey Email: [email protected]

Alexander N. Pisarchik Center for Biomedical Technology Technical University of Madrid Campus Montegancedo 28223 Pozuelo de Alarcon, Madrid, Spain E-mail: [email protected]

Josep J. Masdemont Department of Mathematics. Universitat Politecnica de Catalunya. Diagonal 647 (ETSEIB,UPC) Email: [email protected]

Marian Gidea Department of Mathematical Sciences Yeshiva University New York, NY 10016, USA Fax: +1 212 340 7788 Email: [email protected]

Gennady A. Leonov Department of Mathematics and Mechanics St-Petersburg State University 198504, Russia Email: [email protected]

Edgardo Ugalde Instituto de Fisica Universidad Autonoma de San Luis Potosi Av. Manuel Nava 6, Zona Universitaria San Luis Potosi SLP, CP 78290, Mexico Email: [email protected]

Juan Luis García Guirao Department of Applied Mathematics Technical University of Cartagena Hospital de Marina 30203-Cartagena, SPAIN Fax:+34 968 325694 Email: [email protected]

Elbert E.N. Macau Laboratory for Applied Mathematics and Computing, National Institute for Space Research, Av. dos Astronautas, 1758 C. Postal 515 12227-010 - Sao Jose dos Campos - SP, Brazil Email: [email protected], [email protected]

Michael A. Zaks Institut für Physik Humboldt Universität Berlin Newtonstr. 15, 12489 Berlin Email: [email protected]

Mokhtar Adda-Bedia Laboratoire de Physique Ecole Normale Supérieure de Lyon 46 Allée d’Italie, 69007 Lyon, France Email: [email protected]

Ravi P. Agarwal Department of Mathematics Texas A&M University – Kingsville, Kingsville, TX 78363-8202, USA Email: [email protected]

Editorial Board Vadim S. Anishchenko Department of Physics Saratov State University Astrakhanskaya 83, 410026, Saratov, Russia Fax: (845-2)-51-4549 Email: [email protected]

Continued on the inside back cover

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 6, Issue 1, March 2017

Editors Valentin Afraimovich Xavier Leoncini Lev Ostrovsky Dimitry Volchenkov

L&H Scientific Publishing, LLC, USA

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Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 1–9

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Potential Symmetries, Lie Transformation Groups and Exact Solutions of Kdv-Burgers Equation XiaoMin Wang, Sudao Bilige†, YueXing Bai College of Sciences, Inner Mongolia University of Technology, Hohhot, 010051, China Submission Info Communicated by A.C.J. Luo Received 3 December 2015 Accepted 16 December 2015 Available online 1 April 2017 Keywords Potential symmetry Differential characteristic set algorithm Lie transformation groups KdV-Burgers equation

Abstract In this paper, the classical symmetries and the potential symmetries of KdVBurgers equation are calculated based on differential characteristic set algorithm, and the corresponding Lie transformation groups and invariant solutions of the potential symmetry are derived. Moreover a series of new exact solutions for KdV-Burgers equation are obtained by acting Lie transformation groups on the invariant solutions. It is important that these solutions can not be obtained from the classical symmetries of KdV-Burgers equation.

©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The classical symmetry method, originally developed by the Norwegian mathematician Sophus Lie (1842-1899), leads us to one-parameter groups of transformations acting on the space of independent and dependent variables that leave the considered (partial) differential equations (PDEs) unchanged [1–3]. It is well known that Lie symmetry is the most widely used in solving PDEs, symmetry can be used to construct the analytic solutions of PDEs, the reduction of equations, Hamiltonian structure, Blow-up of solutions and so on. Moreover, this method has had a profound impact on both pure and applied areas of mathematics, physics, and mechanics, etc [4,5]. For PDEs, admitting symmetry is one of the intrinsic nature of the equations. Based on the symmetries of a PDEs, many other important properties of the equation such as Lie algebras [6], conservation laws [7–11], similarity reduction [12, 13], exact solutions [14, 15] can be considered successively. The classical symmetries of many PDEs are insufficient, so it is difficult to solving abundant solutions of PDEs by using the classical symmetry method. Many generalizations of the classical symmetry method have been recently developed [16], such as the potential symmetry [3], the non-classical symmetry [17, 18] etc., where the potential symmetry is clever idea, operation simple and effective expand the classical symmetry [19–20]. Lie’s algorithm, which is the major method with respect to determining symmetries, transforms the problem of determining symmetries into that of determining corresponding infinitesimal vectors whose infinitesimal † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2017.03.001

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XiaoMin Wang, Sudao Bilige, YueXing Bai / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 1–9

functions are found as solutions of some over-determined system of PDEs, called the determining equations (DTEs) [1]. In determining symmetries, tedious, mechanical computations are involved and the order relation of unknown quantities have not been considered in conventional Lie’s algorithm, which result many problems, such as infinite loops on computers, a mass of work and so on. According to the investigations, differential characteristic set algorithm (aslo called Wu’s method) extended and constructed by Temuer Chaolu can partially solve the above-mentioned problems [22]. This algorithm has been successfully applied to classical and nonclassical symmetries [23–27], symmetry classification [27–29], approximate symmetries [30], potential symmetries [19, 31], conservation laws [7–9, 32] of PDEs, which has promoted the investigations of symmetry theory of PDEs. 2 Symmetry and exact solutions of KdV-Burgers equation We consider KdV-Burgers equation ut + uux − α uxx + β uxxx = 0,

(1)

where α , β are constants. 2.1

The classical symmetries of KdV-Burgers equation

The symmetry group of Eq.(1) will be generated by the vector field of the form X =ξ

∂ ∂ ∂ +τ +η , ∂x ∂t ∂u

(2)

where ξ , τ , η are the infinitesimal functions of the symmetry with respect to x,t, u. According to the Lie algorithm, DTEs of the symmetry (2) which is too difficult to get the solutions can be obtained. However, we can obtain the system of equations corresponding to the characteristic set that is equivalent to DTEs by using differential characteristic set algorithm [22]. By solving the above system, we derive the symmetry classification of KdV-Burgers equation as follows Table1. Table 1 The classical symmetry classification of KdV-Burgers equation Parameters α , β

Infinitesimal vector

Arbitrary

X = (at + c) ∂∂x + b ∂∂t + a ∂∂u

α =0 β =0 α =β =0

X = (ax + bt + c) ∂∂x + (d − 3at) ∂∂t + (b − 2au) ∂∂u

X = [(at + b)x + ct + d] ∂∂x + (at 2 + 2bt + k) ∂∂t + (ax − atu − bu + c) ∂∂u X = ξ (x,t, u) ∂∂x + τ (x,t, u) ∂∂t + ϕ (x,t, u) ∂∂u

where ϕt + uϕx = 0, ϕ − ξt − uξx + uτt + u2 τt = 0

where a, b, c, d, k are arbitrary constants. 2.2

The potential symmetries of KdV-Burgers equation

The first conserved form of KdV-Burgers equation (1) is the following form ut + ( 12 u2 − α ux + β uxx )x = 0.

(3)

For obtaining the potential symmetry of KdV-Burgers equation, we introduce the potential variable v, then Eq. (3) is equivalent to the potential system vx = u,

1 vt = −( u2 − α ux + β uxx ). 2

(4) (5)

XiaoMin Wang, Sudao Bilige, YueXing Bai / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 1–9

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The symmetry group of Eqs.(4), (5) will be generated by the vector field of the form X = ξ ∂∂x + τ ∂∂t + η ∂∂u + ϕ ∂∂v ,

(6)

where ξ , τ , η , ϕ are the infinitesimal functions with respect to x,t, u, v. Definition 1. If ξv2 + τv2 + ηv2 = 0, i.e., if and only if the infinitesimal functions ξ , τ , η have an essential dependence on the potential variable v, then X is called the potential symmetry vector of KdV-Burgers equation (1). According to the Lie algorithm and differential characteristic set algorithm, we derive the symmetry classification of Eqs.(4), (5) as follows Table 2. Table 2 The symmetry classification of Eqs.(4), (5) Parameters α , β

Infinitesimal vector

Arbitrary

X = (a1t + a4 ) ∂∂x + a2 ∂∂t + a1 ∂∂u + (a1 xt + a3 ) ∂∂v

X = (a1 x + a2 t + a4 ) ∂∂x + (3a1 t + a5 ) ∂∂t + (−2a1 u + a2 ) ∂∂u + (a2 x − a1 v + a3 ) ∂∂v

α =0

X = (a1 xt + a2 x + a3 t + a4 ) ∂∂x + (a1 t 2 + 2a2 t + a5 ) ∂∂t

+[(h(x,t)ev/2α − (a1 t + a2 ))u + 2α hx (x,t)ev/2α + a1 x + a3 ] ∂∂u

β =0

+[2α h(x,t)ev/2α + a21 x2 + a3 x + α a1 t + a6 ] ∂∂v

where ai (i = 1, . . . , 6) are arbitrary symmetry parameters, and h(x,t) satisfies the linear heat equation ht = α hxx .

(7)

Remark 1. We don’t consider the special case of Eqs.(4), (5) for a = β = 0. In the following, we consider Eqs.(4), (5) for β = 0. We can obtain six finite symmetries and one infinite symmetry of Eqs.(4), (5): 1 ∂ ∂ ∂ ∂ ∂ ∂ ∂ + t 2 + (x − tu) + (α t + x2 ) , X2 = x + 2t − u , ∂x ∂t ∂u 2 ∂v ∂x ∂t ∂u ∂ ∂ ∂ ∂ ∂ ∂ + x , X4 = , X5 = , X6 = , X3 = t + ∂x ∂u ∂v ∂x ∂t ∂v ∂ ∂ X∞ = ev/2α [uh(x,t) + 2α hx (x,t)] + 2α h(x,t)ev/2α . ∂u ∂v X1 = xt

(8)

According to Definition 1, X∞ is the potential symmetry of KdV-Burgers equation (1). Remark 2. The potential symmetry X∞ is given in reference [3] too. It is interesting that a solution of the linear heat equation (7) yields a potential symmetry of Eq.(1). As we know, the heat equation (7) has infinite number of specific known exact solutions. Some representatives of them are listed as follows: h(x,t) = Ax + B, h(x,t) = A(x2 + 2α t) + B, h(x,t) = A(x3 + 6α xt) + B, h(x,t) = A(x4 + 12α x2 t + 12α 2t 2 ) + B, x2 x A h(x,t) = √ e− 4αt + B, h(x,t) = A · erfc( √ ) + B, t 2 αt 2

h(x,t) = Ae−α k t cos(kx + B) +C, h(x,t) = Ae−kx cos(kx − 2α k2t + B) +C,

(9)

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where A, B,C, k are arbitrary constants, erfc(z) is error function ˆ z 2 2 e−y dy. erfc(z) = √ π 0 In general, we can obtain new potential symmetry by choosing h(x,t) which satisfy the relationship (7). Such as, (i) When h(x,t) = 1, X∞ is reduced to the result of reference [33], namely X7 = uev/2α

∂ ∂ + 2α ev/2α . ∂u ∂v

Let X8 = X1 + X7 , X9 = X3 + X7 , X10 = X4 + X5 + X6 + X7 , then 1 ∂ ∂ ∂ ∂ + t 2 + (x − tu + uev/2α ) + (α t + x2 + 2α ev/2α ) , ∂x ∂t ∂u 2 ∂v ∂ v/2α ∂ v/2α ∂ ) + (x + 2α e ) , X9 = t + (1 + ue ∂x ∂u ∂v ∂ ∂ ∂ ∂ + + uev/2α + (1 + 2α ev/2α ) . X10 = ∂x ∂t ∂u ∂v X8 = xt

(ii) When h(x,t) = A(x4 + 12α x2 t + 12α 2 t 2 ) + B, we obtain the potential symmetry X11 = [8α A(x3 + 6α xt) + A(x4 + 12α x2 t + 12α 2 t 2 )u + Bu]ev/2α (iii) When h(x,t) =

2

A − 4xα t √ e t

∂ ∂ + 2α [A(x4 + 12α x2 t + 12α 2 t 2 ) + B]ev/2α . ∂u ∂v

+ B, we obtain the potential symmetry x2

3

3

X12 = (Atu + Be 4αt t 2 u − Ax)t − 2 e

2vt−x2 4α t

x2 A ∂ ∂ + 2α [ √ e− 4αt + B]ev/2α . ∂u ∂v t

(iv) When h(x,t) = Ae−kx cos(kx − 2α k2 t + B) +C, we obtain the potential symmetry v

X13 = [Cuekx + A(u − 2α k) cos(kx − 2α k2 t + B) − 2α Ak sin(kx − 2α k2 t + B)]e 2α −kx +2α [Ae−kx cos(kx − 2α k2 t + B) +C]ev/2α 2.3

∂ . ∂v

∂ ∂u

Lie transformation groups

In the following, we calculate one-parameter Lie transformation groups of the symmetries. Firstly, we can denote the classical symmetries of KdV-Burgers equation (β = 0) from Table 1 X1 = xt ∂∂x + t 2 ∂∂t + (x − tu) ∂∂u , X2 = X2 , X3 = t ∂∂x + ∂∂u , X4 = X4 , X5 = X5 .

(10)

A one-parameter Lie transformation group corresponding to the symmetry X3 satisfies the initial value problem ∗ du∗ dx∗ ∗ dt = t , = 0, = 1, dε ∗ dε ∗ dε ∗ x∗ |ε =0 = x, t ∗ |ε =0 = t, u∗ |ε =0 = u.

Solving (11), we obtain a one-parameter Lie transformation group  x∗ = x + t ε , t ∗ = t, (I) u∗ = u + ε .

(11)

XiaoMin Wang, Sudao Bilige, YueXing Bai / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 1–9

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Similarly, we can determine the one-parameter Lie transformation groups corresponding to the symmetries X9 , X10 , namely ⎧ ⎪ x∗ = x + t ε , t ∗ = t, ⎪ ⎪ √ x2 ⎪ √ ε (2x+t ε ) v v ⎪ ⎪ te 4αt [2α e 2α (e 4α − 1) + t(u + ε )] + απ e 2α (x + t ε )[F(z) − F(z1 )] ⎨ ∗ , u = √ x2 √ v (II) 4α t + 2α (F(z ) − F(z ))] t[ te απ e 1 2 ⎪ ⎪ √ ⎪ ⎪ ⎪ v tε 2 απ x2 ⎪ ⎩v∗ = xε + − 2α ln[e− 2α + √ e− 4αt (F(z) − F(z1 ))], 2 t where F(z) =

´z 0

2

ey dy, z =

√x , 2 αt

z1 =

x+t √ε. 2 αt

⎧ ∗ ∗ ⎪ ⎨x = x + ε , t = t + ε , v ε (III) u∗ = u[1 + 2α e 2α (1 − e 2α )]−1 , ⎪ v+ε ε ⎩ ∗ v = −2α ln[e− 2α − 2α (1 − e− 2α )]. It is easy to determine the one-parameter Lie transformation groups corresponding to the symmetries X11 , X12 , X13 , and these groups are relatively simple. For obtaining abundant exact solutions of KdV-Burgers equation, we determine new Lie transformation groups. Let X14 = X11 + X2 , X15 = X12 + X3 , X16 = X13 + X5 , then we can determine the one-parameter Lie transformation groups corresponding to the symmetries X14 , X15 , X16 , namely ⎧ ∗ ε ∗ 2ε ⎪ ⎪x = e x, t = e t, ⎪ v v ⎪ ⎪ ⎨ ∗ −4u + Aα x(x2 + 6α t)[(e4ε − 1)(e 2α (e4ε − 1)(h(x,t) − B) − 8) + 2Be 2α (1 + e4ε (4ε − 1))] , u = ε [e 2vα ((e4ε − 1)(h(x,t) − B) + 4Bε ) − 4] (IV) e ⎪ ⎪ ⎪ ⎪ ⎪ ⎩v∗ = −2α ln[e− 2vα − 1 (e4ε − 1)(h(x,t) − B) − Bε ], 4 where h(x,t) = A(x4 + 12α x2 t + 12α 2t 2 ) + B. ⎧ ⎪ x∗ = x + t ε , t ∗ = t, ⎪ ⎪ √ x2 +2tv t 2 ε 2 +2xt ε ⎪ √ x2 v v ⎪ ⎪ ⎨ ∗ t 3/2 e 4αt (u + ε ) + 2α B te 4αt (e 4αt − 1) − Aε e 2α (x + t ε ) + B πα (x + t ε )e 2α [F(z) − F(z1 )] , u = √ x2 √ v v (V) 4α t − Ae 2α + B 2α (F(z) − F(z1 ))] t[ te πα e ⎪ ⎪ √ ⎪ ⎪ ⎪ x2 −2tv (x + t ε )2 Aε B πα ⎪ ⎩v∗ = − 2α ln[e 4αt − √ + √ (F(z) − F(z1 ))]. 2t t t ⎧ ⎪ x∗ = x, t ∗ = t + ε , ⎪ ⎨ kx− 2vα +A(cos(kx−2α k2 t+B)−sin(kx−2α k2 t+B))−A(cos(kx−2α k2 (t+ε )+B)−sin(kx−2α k2 (t+ε )+B))] , u∗ = − 2kα [kue (VI) v v 2α k2 ekx− 2α (Cε e 2α −1)+A[sin(kx−2α k2 t+B)−sin(kx−2α k2 (t+ε )+B)] ⎪ ⎪ v ⎩ ∗ 1 v = −2α ln[e− 2α −Cε − 2α k2 Ae−kx (sin(kx − 2α k2t + B) − sin(kx − 2α k2 (t + ε ) + B))]. Similarly, we can determine Lie transformation groups corresponding to the symmetries X11 + X3 , X13 + X4 , but these groups are very complicated, so we omit the calculation results.

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XiaoMin Wang, Sudao Bilige, YueXing Bai / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 1–9

2.4

Exact solutions of KdV-Burgers equation

The characteristic equations for the symmetry X8 are as follows: du dv dx dt = 2 = = . xt t x − tu + uev/2α α t + 12 x2 + 2α ev/2α

(12)

Solving Eqs.(12), we obtain the similarity variables √ √ v x2 −2tv √ x 2α x2 −2tv 2 απ x F(z), σ = 2α [e 4αt t(x − tu) + 2 απ F(z)]. θ = , ς = √ e 4αt (2e 2α − t) − t t t

(13)

By using the standard symmetry reduction method, we can confirm that the solutions of Eqs.(12) are given as follows: √ x2 √ xt f (θ ) + tg(θ ) + 2 απ (x2 − 2α t) − 4α x te 4αt , u(x,t) = √ x2 √ t 2 f (θ ) + 2 απ xtF(z) − 4α t te 4αt √ x2 √ 4α te 4αt − t f (θ ) − 2 απ xF(z) x2 ], v(x,t) = − 2α ln[ 2t 2α t 3/2

(14) (15)

where f (θ ), g(θ ) are the arbitrary function of θ . (14), (15) give the general form for any group-invariant solutions of Eqs.(4), (5). Thus, inserting (14), (15) into Eqs.(4), (5), we obtain f  (θ ) = 0, g(θ ) = −2α f  (θ ).

(16)

Solving Eqs.(16), we get the following results f (θ ) = a + bx, g(θ ) = −2α b,

(17)

where a, b are arbitrary constants. Then we derive the invariance solutions of Eqs.(4), (5) √ x2 √ axt + bx2 − 2α bt + 2 απ (x2 − 2α t)F(z) − 4α x te 4αt , u1 (x,t) = √ x2 √ t(at + bx + 2 απ xF(z) − 4α te 4αt ) √ x2 √ x2 4α te 4αt − (at + bx) − 2 απ xF(z) ], v1 (x,t) = − 2α ln[ 2t 2α t 3/2

(18) (19)

where u1 (x,t) is the exact solution of Eq.(1) which can not obtained by using the classical symmetries of Eq.(1). By acting Lie transformation group (I) on the solution (18) of Eqs.(4), (5), we obtain another solution of Eq. (1) √ (x−t ε )2 √ axt + bx2 − 2α bt − btxε + 2 απ (x2 − 2α t − xt ε )F(z1 ) − 4α x te 4αt . (20) u2 (x,t) = √ (x−t ε )2 √ t[at + bx − bε t + 2 απ (x − t ε )F(z1 ) − 4α te 4αt ] By substituting (20) into Eqs.(4), (5), we obtain v2 (x,t) =

√ (x−t ε )2 √ tε 2 x2 − t 2 ε 2 + 3α ln(t) + − 2α ln[at + bx − bt ε + 2 απ (x − t ε )F(z1 ) − 4α te 4αt ] +C1 , (21) 2 2t

where C1 is arbitrary constant.

XiaoMin Wang, Sudao Bilige, YueXing Bai / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 1–9

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We can get another new solutions by acting Lie transformation group (II) on the solutions (20), (21) of Eqs. (4), (5) √ (x−2t ε )2 √ axt + bx2 − 2α bt − btxε + 2 απ (x2 − 2α t − 2xt ε )F(z2 ) − 4α x te 4αt + G1 (x,t) , u3 (x,t) = √ (x−t ε )2 √ t[at + bx − 2bε t + 2 απ (x − 2t ε )F(z2 ) − 4α te 4αt + G2 (x,t)] √ (x−2t ε )2 √ tε 2 at + bx − 2bt ε − 4α te− 4αt + 2 απ (x − 2t ε )F(z2 ) + G2 (x,t) − 2α ln[ ], v3 (x,t) = xε − 2 3 (x−t ε ) +2C1 t 2 t 2 e 4α t

(22)

(23)

where x2 +2C1 t (x−t ε )2 +2C1 t C1 √ G1 (x,t) = 2α t 3/2 [e 4αt − e 4αt ] + απ txe 2α [F(z1 ) − F(z)], C1 √ x − 2t ε G2 (x,t) = απ te 2α [F(z1 ) − F(z)], z2 = √ . 2 αt

By acting Lie transformation group (III) on the solutions (22), (23) of Eqs.(4), (5), we get the following new solutions G3 (x,t)[b(x − ε )2 + G4 (x,t) + (x − ε )G5 (x,t) + G6 (x,t)]

, (24) ε (1 − e 2α ) + (t − ε )G3 (x,t)[b(x − ε ) + (a − 2bε )(t − ε ) + G4(x,t) + G5 (x,t)] ε a(t − ε ) + b(x − ε − 2t ε + 2ε 2 ) + G4 (x,t) + G5 (x,t) ], (25) v4 (x,t) = −2α ln[−2α (e− 2α − 1) + x2 −2xε +2t(ε +C1 )−ε (ε +2C1 ) 3 4α (t−ε ) (t − ε ) 2 e

u4 (x,t) =

5

2α (t − ε ) 2 e

(x−ε )ε 2α

where G3 (x,t) = e G4 (x,t) = G5 (x,t) = G6 (x,t) = z3 =

(t−ε )2 ε 2 −[x+ε (−1−t+ε )]2 −2C1 (t−ε ) 4α (t−ε )

, [x+ε (−1−2t+2ε )]2 √ √ 2 απ [x + ε (2ε − 1 − 2t)]F(z3 ) − 4α t − ε e 4α (t−ε ) , C1 √ απ (t − ε )e 2α [F(z4 ) − F(z5 )], [x+ε (−1−2t+2ε )]2 √ (t − ε )[a(x − ε ) − 2α b − 2bε (x − ε )] − 4α (x − ε ) t − ε e 4α (t−ε ) √ +2 απ [(x − ε )2 − 2ε (x − ε )(t − ε ) − 2α (t − ε )]F(z3 ), x + ε (−1 − 2t + 2ε ) x + ε (−1 − t + ε ) x−ε   , z4 = , z5 =  . 2 α (t − ε ) 2 α (t − ε ) 2 α (t − ε )

Similarly, by acting Lie transformation group (III) on the solutions (20), (21) of Eqs.(4), (5), we can obtain another new solutions [(t − ε )(a − bε ) + b(x − ε ) + H1(x,t)][b(2α t − x2 ) + a(t − ε )(ε − x) + H2 (x,t)] , (t − ε )[a(ε − t) − bx + bε (1 + t − ε ) − H1(x,t)][H3 (x,t) + H1 (x,t)] ε a(t − ε ) + b(x − ε − t ε + ε 2 ) + H1 (x,t) ], v5 (x,t) = −2α ln[2α (e− 2α − 1) + x2 −2xε +2t(ε +C1 )−ε (ε +2C1 ) 3 4 α (t− ε ) (t − ε ) 2 e

u5 (x,t) =

where [x+ε (−1−t+ε )]2 √ √ H1 (x,t) = 2 απ [x + ε (ε − 1 − t)]F(z4 ) − 4α t − ε e 4α (t−ε ) , [x+ε (−1−t+ε )]2 √ H2 (x,t) = bε [x(2 + t) − 2α − ε (1 + t − ε )] + 4α x t − ε e 4α (t−ε ) √ +2 απ [x2 + xε (ε − 2 − t) + ε (2α + ε − ε 2 ) + t(ε 2 − 2α )]F(z4 ),

3/2

H3 (x,t) = (t − ε )(a − bε ) + b(x − ε ) + 2α (t − ε )

[e

(x−ε )2 +2C1 (t−ε ) 4α (t−ε )

−e

x2 −2xε +2t(ε +C1 )−ε (ε +2C1 ) 4α (t−ε )

].

(26) (27)

8

XiaoMin Wang, Sudao Bilige, YueXing Bai / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 1–9

In addition, we can also continue the above calculation. Such as, by acting Lie transformation groups (IV), (V), (VI) on the above solutions of KdV-Burgers equation, we can obtain more new solutions. Due to the lack of space, we omit the calculation steps. 3 Conclusions Finding more general exact solutions of PDEs have drawn a lot of interests of a diverse group of scientists. It is well known that one generalization of the classical symmetry is the potential symmetry for finding further invariant solutions of PDEs. In this paper, we expanded the classical symmetries of KdV-Burgers equation by the determining potential symmetries, and we obtained a series of new exact solutions of KdV-Burgers equation. Firstly, we determined the classical symmetries and the potential symmetry of KdV-Burgers equation based on differential characteristic set algorithm. The obtained infinite symmetry contained a arbitrary function h(x,t) which satisfies the linear heat equation. By dealing with the linear heat equation, we derived more new potential symmetries. Secondly, we determined one-parameter Lie transformation groups of the potential symmetries. Finally, we obtained the corresponding invariant solutions of the potential symmetry, it is important that these solutions can not be obtained from classical symmetries of the equation. Moreover a series of exact solutions for KdV-Burgers equation are derived by applying Lie transformation group (I), (II), (III) on the invariant solutions (18), (19). Many authors successfully obtained more exact solutions of KdV-Burgers equation by using various methods [34–36]. We can construct the new exact solutions by acting Lie transformation group (I)-(VI) on the solutions obtained by using other methods [34–36]. In our calculation, there are the following diagram I

II

III

III

→ (u2 , v2 ) − → (u3 , v3 ) −→ (u4 , v4 ), (u2 , v2 ) −→ (u5 , v5 ), X8 → (u1 , v1 ) − i

→ denote the action of Lie transformation group i. It is important that these solutions (u j , v j ) can not be where − obtained from the classical symmetries of the equation. In addition, we can also continue the above calculation. Acknowledgments This work is supported by the National Natural Science Foundation of China (11661060, 11571008), Natural Science Foundation of Inner Mongolia Autonomous Region of China (2014MS0114, 2014BS0105), High Education Science Research Program of Inner Mongolia (NJZZ14053). References [1] Lie, S. (1891), Vorlesungen uber Differentialgleichungen Mit Bekannten Infinitesimalen Transformation, Leipzig: BG Teubner. [2] Olver, P.J. (1986), Applications of Lie Groups to Differential Equations, Spinger-Verlag: New York, Inc. [3] Bluman, G. and Kumei, S. (1989), Symmetries and Differential Equations, Spring-Verlag: New York Berlin. [4] Bluman, G., Cheviakov, A., and Anco, S. (2010), Applications of Symmetry Methods to Partial Differential Equations, Spring-Verlag: New York. [5] Ibragimov, N.H. and Ibragimov, R.N. (2012), Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences, Mathematical Modelling of Natural Phenomena, 7, 52-65. [6] Ma, W.X. (1990), K-symmetries and τ -symmetries of evolution equations and their Lie algebras, Journal of Physics A: Mathematical and General, 23, 2707-2716. [7] Bluman, G. and Temuer, Chaolu. (2005), Comparing Symmtries and conservation laws of nonlinear telegraph equations, Journal of Mathematical physics, 46, 1. [8] Bluman, G. and Temuer, Chaolu. (2005), Conservation laws of nonlinear telegraph equations, Journal of Mathematical Analysis and Applications, 310, 459-476.

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[9] Bluman, G. and Temuer, Chaolu. (2006), New conservation laws obtained directly from Symmetry action on a known conservation law, Journal of Mathematical Analysis and Applications, 322, 233-250. [10] Ibragimov, N.H., Ibragimov, R.N., and Galiakberova, L.R. (2014), Symmetries and Conservation Laws of a Spectral Nonlinear Model for Atmospheric Baroclinic Jets, Mathematical Modelling of Natural Phenomena, 9(5):111-118. [11] Avdonina, E.D., Ibragimov, N.H. and Khamitova, R. (2013), Exact solutions of gasdynamic equations obtained by the method of conservation laws, Communications in Nonlinear Science and Numerical Simulation, 18, 2359-2366. [12] Clarkson, P.A. and Kruskal, M.D. (1989), New similarity reductions of the Boussinesq equation, Journal of Mathematical physics, 30, 2201. [13] Lou, S.Y. and Tang, X.Y. (2001), Conditional similarity reduction approach: Jimbo-Miwa equation, Chinese physics B, 10, 897-901. [14] Yang, S.J. and Hua, C.C. (2014), Lie symmetry reductions and exact solutions of a coupled KdV-Burgers equation, Applied Mathematics and Computation, 234, 579-583. [15] Moleleki, L.D. and Khalique, C.M. (2014), Symmetries, Traveling Wave Solutions, and Conservation Laws of a (3+1)Dimensional Boussinesq Equation, Advances in Mathematical Physics, 672679, 1-8. [16] Ibragimov, N.H. (1995), CRC Handbook of Lie Group Analysis of Diefrential Equations: New trends in Theorectieal Developments and Computational Methods, Boca Raton: CRC Press. [17] Rocha, P.M.M., Khanna, F.C., Rocha Filho T.M. and Santana, A.E. (2015), Non-classical symmetries and invariant solutions of non-linear Dirac equations, Communications in Nonlinear Science and Numerical Simulation, 26, 201210. [18] Qian, S.P. and Li, X. (2013), Recursion Operator and Local and Nonlocal Symmetries of a New Modified KdV Equation, Advances in Mathematical Physics, 282390, 1-5. [19] Bluman, G. and Temuer, Chaolu. (2005), Local and nonlocal Symmetries for nonlinear telegraph equation, Journal of Mathematical physics, 46, 023505. [20] Sophocleousa, C. and Wiltshire, R.J. (2006), Linearisation and potential symmetries of certain systems of diffusion equations, Physica A, 370, 329-345. [21] Gandarias, M.L. (2008), New potential symmetries for some evolution equations, Physica A, 387, 2234-2242. [22] Chaolu. (1999), Wuwen-tsum-differential characteristic algorithm of symmetry vectors of partial differential equations (in Chinese), Acta Mathematica Sinica, 19, 326-332. [23] Temuer, Chaolu and Bluman, G. (2014), An algorithmic method for existence of nonclassical symmetries of partial differential equations without solving determining equations, Journal of Mathematical Analysis and Applications, 411, 281-296. [24] Sudao, Bilige, Wang, X.M. and Wuyun, Morigen. (2014), Application of the symmetry classification to the boundary value problem of nonlinear partial differential equations (in Chinese), Acta Physica Sinic, 63, 040201. [25] Lu, L. and Temuer, Chaolu. (2011), A new method for solving boundary value problems for partial differential equations, Computers and Mathematics with Applications, 61, 2164-2167. [26] Eerdunbuhe and Temuer, chaolu. (2012), Approximate solution of the magneto-hydrodynamic flow over a nonlinear stretching sheet, Chinese physics B, 21, 035201. [27] Temuer, Chaolu and Bai, Y.S. (2010), A new algorithmic theory for determining and classifying classical and nonclassical symmetries of partial differential equations (in Chinese), Scientia Sinica Mathematica, 40, 331-348. [28] Chaolu Temuer and Bai, Y.S. (2009), Differential characteristic set algorithm for the complete symmetry classification of partial differential equations, Applied Mathematics and Mechanics (English Edition), 30, 595-606. [29] Temuer Chaolu and Pang, J. (2010), An algorithm for the complete symmetry classification of differential equations based on Wu’s method, Journal of Engineering Mathematics, 66, 181-199. [30] Tumuer Chaolu and Bai, Y.S. (2011), An Algorithm for Determining Approximate Symmetries of Differential Equations Based on Wu’s Method, Chinese Journal of Engineering Mathematics, 28, 617-622. [31] Sudao Bilige and Chaolu. (2006), Calculating potential symmetries and invariant solutions of BBM-Burgers equation by Wu’s method (in Chinese), Journal of Inner Mongolia University (Natural Science Edition), 37, 366-373. [32] Temuer Chaolu, EerDun Buhe and Zheng, L.X. (2007), Auxiliary equation(s) method to determine extended conservation laws and symmetries for partial differential equation(s) and applications of differential Wu’s method (in Chinese), Acta Mathematicae Applicatae Sinica, 30, 910-927. [33] Zhu, Y.P., Ji F.Y., and Chen, X.Y. (2013), Potential symmetries and invariant solutions of generalized KdV-Burgers equation (in Chinise), Pure and Applied Mathematicas, 29, 164-171. [34] Demiray H. (2004), A travelling wave solution to the KdV-Burgers equation, Applied Mathematics and Computation, 154, 665-670. [35] Demiray H. (2005), A complex travelling wave solution to the KdV-Burgers equation, Physics Letters A, 344, 418-422. [36] Soliman, A.A. (2009), Exact solutions of KdV-Burgers equation by Exp-function method, Chaos, Solitons & Fractals, 41, 1034-1039.

Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 11–17

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Conservation Laws in Group Analysis of Gas Filtration Model S.V. Khabirov† Mavlutov’s Institute of Mechanics RAS, 71 October st., Ufa, 450054, Russia Submission Info Communicated by A.C.J. Luo Received 3 December 2015 Accepted 8 March 2016 Available online 1 April 2017 Keywords Potential of conservation law Group properties Optimal system of subalgebra Invariant and patial invariant solutions

Abstract One-dimensional gas filtration was described nonlinear parabolic equation as the conservation law. The potential of the conservation law satisfies a equation as the conservation law. The introduce of the second potential gives a system of equations which admits 6-dimensional Lie algebra. This extends group properties of initial model. With the help of optimal system of subalgebras are classified all invariant and partial invariant solutions which are reduced to invariant solutions of the initial model. Some time it is possible to find integral of invariant submodel.

©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The model of gas filtration in a porous media is obtained from the conservation law of a mass, the state equation and the filtration law. In the one-dimensional case the system of equation is reduced to a nonlinear equation of the parabolic type for the pressure (the equation of piezoconductivity) [1] pt = (ppx )x . The equation of piezoconductivity admits 4-dimensional Lie algebra with the help of which are considered some invariant solutions. This equation is written as the conservation law and therefor one can introduce a potential with the help of which the equation is fulfilled identically. The potential satisfies an equation as the conservation law. Hence one can introduced a new potential. The same competible overdetermined system of equations for 3 functions was obtained. The group properties of this system are extended to 6-dimensional Lie algebra. The optimal system of dissimilar subalgebra classifies all invariant and partial invariant solutions [2, 3]. 2 Group properties of equations with potentials The equation of piezoconductivity is written as the conservation law therefor one can introduced the potential ϕ p = ϕx ,

ppx = ϕt .

† Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2017.03.002

(1)

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S.V. Khabirov / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 11–17

The potential satisfies a equation as the conservation law

ϕt = 2−1 (ϕx2 )x . Hence one can introduced the second potential ψ 2−1 ϕx2 = ψt .

ϕ = ψx ,

(2)

The system (1), (2) has 4 equation for 3 functions. It is compatible obviously. Theorem 1. The Lie algebra L6 admitted by system (1), (2) is 6-dimensional. The basic operators are X1 = ∂x ,

X2 = ∂t ,

X3 = ∂ψ ,

X5 = −t ∂t + p∂ p + ϕ∂ϕ + ψ∂ψ ,

X4 = ∂ϕ + x∂ψ ,

X6 = x∂x + 2p∂ p + 3ϕ∂ϕ + 4ψ∂ψ .

The calculation of admitted algebra is developed by well-known algorithm [3]. Remark 1. The Lie algebra admitted by the piezoconductivity equation is 4-dimensional. The Lie algebra admitted by system (1) is 5-dimensional. Nontrivial commutators of the basic operators are [X1 , X4 ] = X3 ,

[X1 , X6 ] = X1 ,

[X3 , X6 ] = 4X3 ,

[X2 , X5 ] = −X2 ,

[X4 , X5 ] = X4 ,

[X3 , X5 ] = X3 ,

[X4 , X6 ] = 3X4 .

The algebra L6 is decomposed in semidirect sum of ideal J = {X1 , X2 , X3 , X4 } and Abelian subalgebra A = {X5 , X6 } ˙ (3) L6 = J ⊕A. The inner automorphysms of algebra L6 are calculated by rule Xa i = [Xi , X  ],

X  |ai =0 = X = x j X j .

The result of calculations consists from 6 automorphisms (the transformations of invariable variables are absent) A1 : x1 = x6 a1 + x1 ,

x3 = x4 a1 + x3 ;

A2 : x2 = −x5 a2 + x2 ; A3 : x3 = (x5 + 4x6 )a3 + x3 ;

A4 : x3 = −x1 a4 + x3 ,

x4 = (x5 + 3x6 )a4 + x4 ;

A5 : x2 = x2 a5 ,

x3 = x3 a−1 5 ,

A6 : x1 = x1 a6 ,

x3 = x3 a46 ,

x4 = x4 a−1 5 ; x4 = x4 a26 .

The automorphisms A5 , A6 are identical at a5 = a6 = 1. There are discrete automorphisms for L6 D1 : x1 → −x1 ,

x4 → −x4 ;

D3 : x3 → −x3 ,

D2 : x2 → −x2 ;

x4 → −x4 .

The optimal system of Abilian subalgebra A consists from subalgebras 0,

X5 + aX6 ,

{X5 , X6 }.

(4)

S.V. Khabirov / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 11–17

13

The optimal system for L6 is constructed with the help of decomposition (3) according to rule. Let subalgebra with given dimension has the projection in A from the list (4). Adding to basic operators of projection an arbitrary linear combination of the basic elements of ideal we obtain the representation of a subalgebra from L6 . With the help of automorphisms some coefficients of the linear combination are equal to 0 or 1. At last the conditions of subalgebra are verified: the commutator of basic elements must be linear combination of the basic operators. Calculations lead to following result. One-dimensional subalgebras (ε =0 or 1, δ =0 or 1) X1 + aX2 + X4 , X1 + ε X2, ε X2 + X4, X2 + X3 , X2 , X3 , X5 + aX6 , X5 + X1 , X5 − 4−1 X6 + X3 , X5 − 3−1 X6 + X4, X6 + ε X2 . 2-dimensional subalgebras {X1 + X4 , X2 + aX3 }, {X1 + aX2 + X4 , X3 }, {X1 , X2 + aX3 }, {X1 + ε X2 , X3 }, {X4 , X2 + aX3 }, {X4 + ε X2 , X3 }, {X2 , X3 } {X1 + X4 , X5 − 2−1 X6 }, {X1 , X5 + aX6 }, {X1 , X5 − 4−1 X6 + X3 }, {X1 + X2 , X5 − X6 }, {X4 , X5 + aX6 }, {X4 , X5 − 4−1 X6 + X3 }, {X2 + X4 , X5 − 2/3X6 }, {X3 , X5 + aX6 }, {X3 , X5 + X1 }, {X3 , X5 − 3−1 X6 + X4 }, {X2 , X5 − 4−1 X6 + X3 }, {X2 , X5 + aX6 }, {X2 , X5 + X1}, {X2 , X5 − 3−1 X6 + X4 }, {X1 , X6 + ε X2}, {X2 , X6 }, {X3 , X6 + ε X2}, {X4 , X6 + ε X2}, {X5 , X6 }. 3-dimensional subalgebras {X1 + X4 , X2 , X3 }, {X1 , X2 , X3 }, {X1 + ε X2, X3 , X4 }, {X1 + δ X4 , X3 , X2 + ε X4}, {X2 , X3 , X4 }, {X1 , X5 , X6 }, {X2 , X5 , X6 }, {X3 , X5 , X6 }, {X4 , X5 , X6 }, {X1 , X2 , X5 + aX6 }, {X1 , X3 , X5 + aX6 }, {X1 , X2 + X3 , 2X5 − X6 }, {X1 + X4 , X2 + aX3 , 2X5 − X6 }, {X1 + X4 , X3 , 2X5 − X6 }, {X1 , X2 , X5 − 4−1 X6 + X3 }, {X2 , X4 , X5 + aX6 }, {X3 , X4 , X5 + aX6 }, {X2 + X3 , X4 , 2X5 − X6 }, {X2 , X4 , X5 − 4−1 X6 + X3 }, {X2 + X4 , X3 , 3X5 − 2X6 }, {X2 , X3 , X5 + aX6 }, {X1 + X2 , X3 , X5 }, {X1 , X3 , X5 + aX6 }, {X2 , X3 , X5 + X1 }, {X1 , X3 , X5 − 3−1 X6 + X4 }, {X2 , X3 , X5 − 3−1 X6 + X4 }, {X1 + X3 , X2 , 3X5 − X6 }, {X2 , X3 + X4 , X5 }, {2X1 + 3X3 , X2 , X5 − 3−1 X6 + X4}, {X1 , X2 , X6 }, {X1 , X3 , X6 + ε X2}, {X2 , X4 , X6 }, {X3 , X4 , X6 + ε X2}, {X2 , X3 , X6 }. 4-dimensional subalgebras {X1 , X2 , X3 , X4 }, {X1 , X2 , X5 , X6 }, {X1 , X3 , X5 , X6 }, {X2 , X3 , X5 , X6 }, {X2 , X4 , X5 , X6 }, {X3 , X4 , X5 , X6 }, {X1 , X2 , X3 , X5 + aX6 }, {X1 , X3 , X4 , X5 + aX6 }, {X2 , X3 , X4 , X5 + aX6 }, {X1 , X2 + X4 , X3 , 3X5 − 2X6 }, {X1 + X2 , X3 , X4 , X5 − X6 }, {X1 + X4 , X2 , X3 , 2X5 − 2X6 }, {X2 , X3 , X4 , X5 + X1 }, {X1 , X2 , X3 , X5 − 3−1 X6 + X4 }, {X1 , X2 , X3 , X6 }, {X1 , X3 , X4 , X6 + ε X2 }. 5-dimensional subalgebras {X1 , X2 , X3 , X4 , X5 + aX6 }, {X1 , X2 , X3 , X4 , X6 }, {X1 , X2 , X3 , X5 , X6 }, {X1 , X3 , X4 , X5 , X6 }, {X2 , X3 , X4 , X5 , X6 }. 3 Invariant solutions Only one-dimensional subalgebras give invariant solutions. At fist invariants of subalgebra must be calculated. The invariants containing dependent variables are assigned as new invariant functions on a invariant containing only independent variables. From here the dependent variables may be expressed through the invariant functions on one variable. That representation of a solution is substituted in the system (1), (2). The invariant submodel from ordinary differential equations for the invariant functions is obtained [3].

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Theorem 2. Invariant submodels of the system (1), (2) coincide with invariant submodels of the system (1) constructed by one-dimensional subalgebras of 5-dimensional Lie algebra. The proof is direct calculating of submodels. As example the invariant solutions on subalgebra X5 − 4−1 X6 + X3 are examined. The representation of this solution has the form p = |t|−1/2 P(s),

ϕ = |t|−1/4 Φ(s),

ψ = − ln |t| + Ψ(s),

s = x|t|−1/4 ,

where s, P, Φ, Ψ are invariants of subalgebra. The substitution of the representation in the system (1), (2) gives a system of ordinary differential equations P = Φ , Φ = Ψ ,

PP = −4−1 (Φ + sΦ ),

(5)

2−1 P2 + 1 + 4−1 sΨ = 0,

(6)

From the equation (5) after integrating the second equation and excluding function Φ it follows the Abilian equation for the function P (7) 4sPP = 2P2 − s2 P −C, where C is a constant. The solutions of the equation (7) are invariant solutions of the system (1) which are obtained if in (5) to exclude function Φ. In our case from (6) it follows C = −4. For C = 0 the solution is p = −6−1 x2 t −1 + Dx|x|−1/2 |t|−5/8 . discribing the spread of a pressure hillock on the interval [0, x1 ). For x = 0 the pressure is zero and the filtration velocity is infinity (the suction from pore). The another end of the interval is moving by the law x1 ∼ |t|1/4 to the domain with vanishing pressure and damping filtration velocity px |x=x1 ∼ |t|−3/4 . The maximal pressure pmax ∼ |t|−1/2 is moving by the law xmax ∼ |t|1/4 . The equation (7) for C = −4 has not singular points. After substitution τ = s2 the equation becomes 2P2 + 4 = τ P(1 + 8Pτ ). The picture of integral curves in the first quadrant may be presented in the following way (Fig.1). The −1 maximum points of integral curves √ lie √on the curve m : τ = 2P + 4P with asymptotes τ = 2P, P = 0 and with the minimum point (P1 , τ1 ) = ( 2, 4 2). The points of inflections for integral curves lie on the curve l :

τ =2 P

(P2 + 2)(3P2 + 2) . P(2 − P2 ) m

P1

ι τ

m

τ

τ

1

Fig. 1

S.V. Khabirov / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 11–17

15

√ with asimptotes P = 0 and P = 2 = P1 . The curve l lies above the curve m. The axis τ consists from points sm of minimums for integral curves τ = τ (P). When τ → ∞ the integral curve P = P(τ ) decrease monotonically, is convex downwards, lie above m and tends to zero P → 0, otherwise if P → P0 = 0 then Pτ → 0, but from equation it follows Pτ → −8−1 contradiction. Hence the integral curves approximate to m as τ → ∞. The straight line √ τ = 0 is integral. The point sm = τm and the point of maximum for integral curve P = P(τ ) tends to zero as t → ∞. The filtration velocity is infinity in the point sm (the suction from pore). The filtration velocity goes to zero when τ → ∞. Thus the integral curve describes damping of the pressure hillock under the suction from pore. 4 Partial invariant solutions The 2-dimensional subalgebra has 3 point invariants. There is a invariant depending only independent variables for some subalgebra. In this case one can consider regular partial invariant solution of rank 1 and defect 1 [2]. If all invariant contain functions then one can consider irregular partial invariant solution of rank 1 and defect 1 which is called usually a simple wave [2]. Theorem 3. The partial invariant solution on 2-dimensional subalgebra is reduced to invariant solution on one-dimensional subalgebra. Proof is developed by calculating invariants of 2-dimensional subalgebras, by substitution of the representation to the system (1), (2) and by finding all derivatives of functions in general situation. By theorem of reduction [2, p.290] there is one-dimensional subalgebra for which the solution should be invariant. We consider 3 examples of reduction. The subalgebra X4 , X5 − 4−1 X6 + X3 gives the representation of regular partial invariant solutions of rank 1 and defect 1 (8) p = |t|−1/2 P(s), ψ = xϕ + X (s) − ln|t|, s = x|t|−1/4 , where P, X , s are invariants of subalgebra. The substitution of representation into system (1) lead to relations

ϕx = |t|−1/2 P(s),

ϕt = |t|−5/4 PP .

In a new independent variables t, s the relations have the form

ϕs = |t|−1/4 P(s),

ϕt = |t|−5/4 P(P + 4−1 s),

which to within the Galilei transformation (X4 ) have the integral

ϕ = −4|t|−1/4 P(s)(P + 4−1 s), where the function P satisfies the equation PP + P2 + 4−1 sP + 2−1 P = 0.

(9)

This solution is invariant with respect to subalgebra X5 − 4−1 X6 + X3 . The substitution of representation (8) into system (2) gives the relation (7) with C = −4 X  + sP = 0,

PP + 4−1 sP = s−1 (2−1 P2 + 1).

(10)

The last equation in (10) is the partial integral for (9). The subalgebra {X4 , X5 + aX6 } gives the representation of regular partial invariant solution of rank 1 and defect 1 (11) p = |t|−1−2a P(s), ψ = xϕ + |t|−1−4a X (s), s = x|t|a ,

16

S.V. Khabirov / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 11–17

where P, X , s are invariants of subalgebra. The substitution of representation (11) into the system (1) lead to relations ϕx = |t|−1−2a P, ϕt = |t|−2−3a PP. In a new independent variables t, s the relations are written

ϕs = |t|−1−3a P,

ϕt = |t|−2−3a P(P − asσ ),

σ = t|t|−1 .

which to within the Galilei transformation (X4 ) are integrating

ϕ = |t|−1−3a (1 + 3a)−1 P(as − Pσ ), where the function P satisfies the equation

σ (PP + P2 ) = asP − (1 + 2a)P.

(12)

There is reduction to the invariant solution of subalgebra X5 + aX6 . The substitution of representation (11) into system (2) gives the relations from which follows equation (12) also. If a = −3−1 then the equation (12) has integral σ PP = C − 3−1 sP. It is the Abilian equation without singular points (C = 0). As σ = 1 (t > 0), C > 0 the curve m of maximum points for integral curves is a hyperbola P = 3Cs−1 , the curve l of inflection points for integral curves is s = 3CP−1 +CP2 . The axis s consists from maximum points for integral curves s = s(P). The picture of the integral curves (Fig.2) describes damping of the pressure hillocks with zero filtration in infinity and infinity filtration in the moving points x = sk t 1/3 (sk = 0 is the stationary pore).

P

ι m

s

s

K

Fig. 2

The subalgebra {X2 , X5 + X1 } gives the representation of irregular partial invariant solution of rank 1 and defect 1 (13) p = ex P(α ), ϕ = ex Φ (α ), ψ = ex Ψ(α ), where P, Φ , Ψ are invariants of subalgebra and α (t, x) is an arbitrary function. The substitution of representation (13) into (1), (2) gives relations

αx =

P−Φ 1 Φ −Ψ , = =    Φ Ψ K (α )

e−x αt =

P2 P2 PP == + = L(α ). 2Ψ Ψ Φ  K 

From here it follow equations K(α ) = x + ω (t),

K  LeK = ω  E ω .

S.V. Khabirov / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 11–17

17

In the last equation the variables α and t are separated and its should be new independent variables. Hence eω = Ct and the function α depend on only one veriable s = x+ ln |t|. The representation (13) should be rewritten p = t −1 P(s),

ϕ = t −1 Φ (s),

ψ = t −1 Ψ(s).

(14)

It is the representation of the invariant solution for sabalgebra X1 + X5 . The substitution of representation (14) into (1), (2) gives the system P = Φ , which is equivalent to one equation

PP = −Φ + Φ  ,

Φ = Ψ ,

PP + P2 = P − P.

2−1 P2 = −Ψ + Ψ , (15)

The stationary point of an integral curve of the equation (15) P = Pm > 0, P = 0 is the point of maximum = −1. If P → P0 = 0 as s → ∞ then P → 0, P → −1 and it follows contradiction. Hence P0 = 0. The integral curve starting from origin P = s = 0 has the series expansion P = s − 4−1 s2 − 17−1 s3 − ... This series describes the bow of pressure hillock with the finite filtration velocity under t = 1. As t → ∞ the bow x → −∞ and the pressure vanishes p = t −1 P → 0. As x → ∞ the filtration is absent. Thus the solutions of equation (15) describe runing damping hillocks of pressure with the finite filtration velocity in the bow where the pressure is equal zero. P

Remark 2. For subalgebras of the dimensions 3, 4, 5 all partial invariant and differential invariant submodels [4] of rank 1 are reduced to invariant solutions. Remark 3. Differential invariant submodels of rank 2 give invariant differential substitutions which are determined by differential invariants. 5 Conclusions The introduce of 2 potentials for the equation of one-dimensional gas filtration extends the group properties on 2 dimensions of admited Lie algebra. But invariant and partial invariant submodels are reduced to invariant submodels of the initial equation. Novelty of the research is in the finding of the integrals of the invariant submodels. Acknowledgements This research was supported by the RFFI (14-01-97027-p-a) and the government of RF decree 220 grat 11.G34.31.0042. References [1] Barenblut, G.I., Entov, V.M., and Rizhik, V.M. (1972), Theory of unstable filtration of fluid and gas, Nedra: Moscow. [2] Ovsiannikov, L.V. (1978), Group Analysis of Differential Equations, Nauka: Moscow, (English translation, Ames, W.F.(ed.), Academic Press: New York, 1982). [3] Chirkunov, Yu.A., Khabirov, S.V. (2012), Elements of symmetry analysis of differential equations of continuum mechanics, NGTU: Novosibirsk. [4] Khabirov, S.V. (2013), A hierarchy of submodels of differential equations, Siberian Mathematical Journal, 54(6), 1111-1120.

Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 19–34

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Existence of Semi Linear Impulsive Neutral Evolution Inclusions with Infinite Delay in Frechet Spaces Dimplekumar N. Chalishajar†, K. Karthikeyan ‡, A. Anguraj § Department of Applied Mathematics, Virginia Military Institute (VMI), 431, Mallory Hall, Lexington, VA 24450, USA Department of Mathematics, KSR College of Technology, Tiruchengode, Tamil Nadu 637215, India Department of Mathematics, PSG College of Arts and Science, Coimbatore, Tamil Nadu 641 014, India Submission Info Communicated by A.C.J. Luo Received 17 September 2015 Accepted 8 April 2016 Available online 1 April 2017 Keywords Impulsive differential inclusions Fixed point Frechet spaces Nonlinear alternative due to Frigon

Abstract In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We study a new phase space for impulsive system with infinite delay.

©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In recent years, impulsive differential and partial differential equations have become important in mathematical models of real phenomena relating to biological and medical domains. In these models, the investigated simulating processes and phenomena are often subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. The theory of impulsive differential equations has seen considerable development, see the monographs of Bainov and Semeonov [1], Lakshimikantham et al. [2] and Perestyuk [3]. Simultaneously the theory of impulsive differential equations as much as neutral differential equations has been emerging as an important area of investigations in recent years, stimulated by their numerous applications to problems in physics, mechanics, electrical engineering, medicine biology, ecology, and so on. The impulsive differential systems can be used to model processes which are subject to abrupt changes, † Corresponding

author. Email address: [email protected], [email protected] ‡ Email address: karthi [email protected] § Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2017.03.003

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Dimplekumar N. Chalishajar, K. Karthikeyan, A. Anguraj / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 19–34

and which cannot be described by the classical differential systems, we refer to the readers [2]. Partial neutral integro-differential equation with infinite delay has been used for modeling the evolution of physical systems, in which the response of the system depends not only on the current state, but also on the past history of the system, for instance, for the description of heat conduction in materials with fading memory, we refer to the papers of Gurtin and Pipkin [4], Nunziato [5], and the references therein related to this matter. In [6] authors are concerned with the observer design problem for a class of linear delay systems of the neutral-type. The problem addressed is that of designing a full-order observer that guarantees the exponential sta- bility of the error dynamic system. In [7] authors studied the problem of cost-guaranteed dynamic output feedback control for a class of continuous-time linear switched system with both discrete and neutral delays with impulse term. A sufficient condition is studied in terms of a set of linear matrix inequalities, to guarantee the exponential stability and a certain bound for the cost function of the closed-loop system, where the decay estimate is explicitly given to quantify the convergence rate. They used the average dwell time approach and the piecewise Lyapunov function technique. Recently, several works reported existence results for mild solutions for impulsive neutral functional differential equations or inclusions, such as [8, 9] and references therein. However, the results obtained are only in connection with finite delay. Since many systems arising from realistic models heavily depend on histories (i.e., there is an effect of infinite delay on state equations), there is a real need to discuss partial functional differential systems with infinite delay, where numerous properties of their solutions are studied and detailed bibliographies are given. The literature related to first and second order nonlinear non autonomous neutral impulsive systems with or without state dependent delay is not vast. To the best of our knowledge, this has not been thoroughly explored and is one of the main motivations of this paper. When the delay is infinite, the notion of phase space plays an important role in the study of both qualitative and quantitative theory. A common choice is a semi-normed space satisfying suitable axioms, introduced by Hale and Kato in [10]; see also Corduneanu and Lakshmikantham [11]; J. R. Graef [12] and S. Baghli and M. Benchohra [13, 14]. Unfortunately, we have not discovered a detailed treatment of a system involving infinite delay with impulse effects. Henderson and Ouahab [15] discussed existence results for non-densely defined semi-linear functional differential inclusions in Frechet spaces. Hernandz et al. [16] studied existence of solutions for impulsive partial neutral functional differential equations for first and second order systems with infinite delay. Recently, Arthi and Balachandran [17] proved controllability of the second order impulsive functional differential equations with state dependent delay using a fixed point approach and a cosine operator theory. It has been observed that the existence or controllability results proved by different authors are based on an axiomatic definition of the phase space given by Hale and Kato [10]. However, as remarked by Hino, Murakami, and Naito [18], it has come to our attention that these axioms for the phase space are not correct for the impulsive system with infinite delay (refer [19, 20]). This motivated us to generate a new phase space for the existence of a non-autonomous impulsive neutral inclusion with infinite delay. This direction is another focus of our paper and to the best of our knowledge, has not yet been considered in the literature. On the other hand, researchers have proved the controllability results using the compactness assumption of semigroups and the family of cosine operators. However, as remarked by R. Triggiani [21], if X is an infinite dimensional Banach space, then the linear control system is never exactly controllable on given interval if either B is compact or associated semigroup is compact. According to R. Triggiani [21], this is a typical case for most control systems governed by parabolic partial differential equations and hence the concept of exact controllability is very limited for many parabolic partial differential equations. Nowadays, researchers are driven to overcome this problem, refer to ( [17, 19, 20]). Very recently, Chalishajar and Acharya [20] studied the controllability of second order neutral functional differential inclusion, with infinite delay and impulse effect on unbounded domain, without compactness of the family of cosine operators. Ntouyas and O’Regan [22] gave some remarks on controllability of evolution equations in Banach paces and proved a result without compactness assumption. The rest of this paper is organized as follows: In Section 2 we introduce the system, recall some basic

Dimplekumar N. Chalishajar, K. Karthikeyan, A. Anguraj / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 19–34

21

definitions, and preliminary facts which will be used throughout this paper. The existence theorems for semi linear impulsive neutral evolution inclusions with infinite delay, and their proofs are arranged in Section 3. Finally, in Section 4, an example is presented to illustrate the applications of the obtained results. 2 Preliminaries In this paper, we shall consider the existence of mild solutions for first order impulsive partial neutral functional evolution differential inclusions with infinite delay in a Banach space E d [y(t) − g(t, yt )] ∈ A(t)y(t) + F(t, yt ) dt t ∈ J = [0, +∞), t = tk , k = 1, 2, . . . Δy|t=tk = Ik (y(tk− )),

k = 1, 2, . . .

y0 = φ ∈ Bh .

(1) (2) (3)

where F : J × Bh → P(E) is a multivalued map with nonempty compact values, P(E) is the family of all subsets of E, g : J × Bh → E and Ik : E → E, k = 1, 2, . . . are given functions, φ ∈ Bh are given functions and {A(t)}0≤t 0, φ (θ ) is bounded and measurable ˆ 0 h(s) sups≤θ ≤0|φ (θ )|ds < +∞}. function on [−r, 0] and −∞

Here, Bh endowed with the norm

ˆ

φ Bh =

0

−∞

h(s) sup |φ (θ )|ds, ∀φ ∈ Bh . s≤θ ≤0

Then it is easy to show that (Bh , .Bh ) is a Banach space. Lemma 1. Suppose y ∈ Bh ; then, for each t > 0, yt ∈ Bh . Moreover, l|y(t)| ≤ yt Bh ≤ l sup |y(s)| + y0 Bh , 0≤s≤t

where l :=

´0

−∞ h(s)ds

< +∞.

Proof. For any t ∈ [0, a], it is easy to see that, yt is bounded and measurable on [−a, 0] for a > 0, and ˆ 0 h(s) sup |yt (θ )|ds yt Bh = ˆ = ˆ =

−∞

−t −∞ −t −∞

θ ∈[s,0]

ˆ

h(s) sup |y(t + θ )|ds + θ ∈[s,0]

h(s) sup |y(θ1 )|ds + θ1 ∈[t+s,t]

ˆ

0 −t 0

−t

h(s) sup |y(t + θ )|ds θ ∈[s,0]

h(s) sup |y(θ1 )|ds. θ1 ∈[t+s,t]

22

Dimplekumar N. Chalishajar, K. Karthikeyan, A. Anguraj / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 19–34

ˆ ≤ ˆ = ˆ ≤ ˆ ≤ ˆ =

−t −∞ −t −∞ −t −∞ 0 −∞ 0 −∞

ˆ h(s)g[ sup

θ1 ∈[t+s,0]

h(s)

sup θ1 ∈[t+s,0]

|y(θ1 )| + sup |y(θ1 )|g]ds +

−t

θ1 ∈[0,t]

ˆ

|y(θ1 )|ds +

0 −∞

0

h(s) sup |y(θ1 )|ds θ1 ∈[0,t]

h(s)ds. sup |y(s)| s∈[0,t]

h(s) sup |y(θ1 )|ds + l. sup |y(s)| θ1 ∈[s,0]

s∈[0,t]

h(s) sup |y(θ1 )|ds + l sup |y(s)| θ1 ∈[s,0]

s∈[0,t]

h(s) sup |y0 (θ1 )|ds + l sup |y(s)| θ1 ∈[s,0]

s∈[0,t]

= l sup |y(s)| + y0 . s∈[0,t]

Since φ ∈ Bh , then yt ∈ Bh . Moreover, ˆ 0 ˆ h(s) sup |yt (θ )|ds ≥ |yt (θ )| yt Bh = −∞

θ ∈[s,0]

0 −∞

h(s)ds = l|y(t)|.

The proof is complete. Above definition of phase space also satisfies the conditions given by Hale and Kato [10]. (A1) if x : (−∞, b] → X , b > 0, continuous on [0, b] and x0 ∈ Bh , then for every t ∈ [0, b] the following conditions hold: (a) xt is in Bh (b) x(t) ≤ Hxt Bh (c) xt Bh ≤ M(t)x0 Bh + K(t) sup {x(s) : 0 ≤ s ≤ t}, where H > 0 is a constant; K, M : [0, ∞) → [1, ∞), K is continuous, M is bounded and H, K, M are independent of x(·). (A2) For the functions x in (A1), xt is Bh valued continuous functions on [0, b]. (A3) The space Bh is complete. Next, we introduce definitions, notation and preliminary facts from multi-valued analysis, which are useful for the development of this paper (refer [23]). Let C([0, b], E) denote the Banach space of all continuous functions from [0, b] into E with the norm y∞ = sup{y(t) : 0 ≤ t ≤ b}. and let L1 ([0, ∞), E) be the Banach space of measurable functions y : [0, ∞) → E, that are Lebesgue integrable with the norm ˆ ∞ y(t)dt for all y ∈ L1 ([0, ∞), E). yL1 = 0

Let X be a Frechet space with a family of semi-norms { · n }n∈N . Let Y ⊂ X , we say that Y is bounded if for every n ∈ N, there exists M¯n > 0 such that yn ≤ M¯n

for all

y ∈ Y.

To X we associate a sequence of Banach spaces {(X n ,  · n )} as follows: For every n ∈ N, we consider the equivalence relation ∼n defined by :x ∼n y if and only if x − yn = 0 for all x, y ∈ X . We denote X n =

Dimplekumar N. Chalishajar, K. Karthikeyan, A. Anguraj / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 19–34

23

(X |∼n ,  · n ) the quotient space, the completion of X n with respect to  · n . To every Y ⊂ X , we associate a sequence {Y n } of subsets Y n ⊂ X n as follows: For every x ∈ X , we denote [x]n the equivalence class of x of subset X n and we define Y n = {[x]n : x ∈ Y }. We denote Y¯n , int(Y n ) and ∂nY n , respectively, the closure, the interior, and the boundary of Y n with respect to  ·  in X n . We assume that the family of semi-norms { · n } verifies: x1 ≤ x2 ≤ x3 ≤ . . .

x ∈ X.

for every

Let (X , d) be a metric space. We use the following notations: Pcl (X ) := {Y ∈ P(X ) : Y closed}, Pb (X ) := {Y ∈ P(X ) : Y bounded} Pcv (X ) := {Y ∈ P(X ) : Y convex}, Pcp (X ) := {Y ∈ P(X ) : Y compact}. Consider Hd : P(X ) × P(X ) → R+ ∪ {∞}, given by Hd (A , B) := max{ sup d(a, B), sup d(A , b)}, a∈A

b∈B

where d(A , b) := infa∈A d(a, b), d(a, B) := inf b∈B d(a, b). Then (Pb,cl (X ), Hd ) is a metric space and (Pcl (X ), Hd ) is a generalized (complete) metric space (see [24]). Definition 1. We say that a family {A(t)}t≥0 generates a unique linear evolution system {U (t, s)}(t,s)∈Δ for Δ1 = {(t, s) ∈ J × J : 0 ≤ s ≤ t < +∞} satisfying the following properties: (1) U (t,t) = I where I is the identity operator in E, (2) U (t, s)U (s, τ ) = U (t, τ ) for 0 ≤ τ ≤ s ≤ t < +∞, (3) U (t, s) ∈ B(E), the space of bounded linear operators on E, where for every (t, s) ∈ Δ1 and for each y ∈ E, the mapping (t, s) → U (t, s)y is continuous. More details on evolution systems and their properties could be found in the books of Ahmed [25], Engel and Nagel [26], and Pazy [27]. Definition 2. A multivalued map G : J → Pcl (X ) is said to be measurable if for each x ∈ E, the function Y : J → X defined by Y (t) = d(x, G(t)) = inf{|x − z| : z ∈ G(t)}. is measurable where d is the metric induced by the normed Banach space X . Definition 3. A function F : J × Bh → P(X ) is said to be an L1loc -Caratheodory multivalued map if it satisfies: (i) x → F(t, y) is continuous(with respect to the metric Hd ) for almost all t ∈ J; (ii) t → F(t, y) is measurable for each y ∈ Bh ; (iii) for every positive constant k there exists hk ∈ L1loc (J; R+ ) such that F(t, y) ≤ hk (t)

for all

yBh ≤ k

and for almost all t ∈ J.

A multivalued map G : X → P(X ) has convex(closed) values if G(x) is convex(closed) for all x ∈ X . We say that G is bounded on bounded sets if G(B) is bounded in X for each bounded set B of X , i.e., sup{sup{y : y ∈ G(x)}} < ∞. x∈B

Finally, we say that G has fixed point if there exists x ∈ X such that x ∈ G(x). For each y ∈ B∗ , let the set SF,y known as the set of selectors from F defined by SF,y = {v ∈ L1 (J; E) : v(t) ∈ F(t, yt ),

a.e.t ∈ J}.

For more details on multivalued maps we refer to the books of J. P. Aubin and A. Cellina [28], Deimling [29], Gorniewicz [30], Hu and Papageorgiou [31], and Tolstonogov [32].

24

Dimplekumar N. Chalishajar, K. Karthikeyan, A. Anguraj / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 19–34

Definition 4. A multivalued map F : X → P(X ) is called an admissible contraction with constant {kn }n∈N if for each n ∈ N there exists kn ∈ (0, 1) such that (i) Hd (F(x), F (y)) ≤ kn x − yn for all x, y ∈ X . (ii) For every x ∈ X and every ε ∈ (0, ∞)n , there exists y ∈ F(x) such that x − yn ≤ x − F(x)n + εn

for every

n ∈ N.

The following nonlinear alternative will be used to prove our main result. Theorem 2. [Nonlinear Alternative of Frigon, [33,34])]. Let X be a Frechet space and U an open neighborhood of the origin in X and let N : U¯ → P(X ) be an admissible multivalued contraction. Assume that N is bounded. Then one of the following statements holds: (C1) N has a fixed point; 2 (C2) There exists λ ∈ [0, 1) and x ∈ ∂ U such that x ∈ λ N(x). 3 Existence results We consider the space PC = {y : (−∞, ∞) → E|y(tk− ) and y(t) = φ (t)

y(tk− ) exist with y(tk ) = y(tk− ),

for t ∈ (−∞, ∞),

yk ∈ C(Jk , E), k = 1, 2, 3, . . . }

where yk is the restriction of y to Jk = (tk ,tk + 1], k = 1, 2, 3, . . . Now we set B∗ = {y : (−∞, ∞) → E : y ∈ PC ∩ Bh }. Bk = {y ∈ B∗ : sup |y(t)| < ∞}, t∈Jk∗

where

Jk∗ = (−∞,tk ].

Let  · k be the semi-norm in Bk defined by yk = y0 Bh + sup{|y(s)| : 0 ≤ s ≤ tk },

y ∈ Bk .

To prove our existence result for the impulsive neutral functional differential evolution problem with infinite delay (1) − (3), firstly we define the mild solution. Definition 5. We say that the function y(·) : (−∞, +∞) → E is a mild solution of the evolution system (1) − (3) if y(t) = φ (t) for all t ∈ (−∞, 0], Δy|t=tk = Ik (y(tk− )), k = 1, 2, . . . and the restriction of y(·) to the interval J is continuous and there exists f (·) ∈ L1 (J; E) : f (t) ∈ F(t, yt ) a.e in J such that y satisfies the following integral equation: ˆ t (4) y(t) = U (t, 0)[φ (0) − g(0, φ )] + g(t, yt ) + U (t, s)A(s)g(s, ys )ds 0 ˆ t + U (t, s) f (s)ds + ∑ U (t,tk )Ik (y(tk− )), for each t ∈ [0, +∞). 0

0 0 such that A−1 (t)B(E) ≤ M0 (H5) There exists constant dk > 0,

k = 1, 2, . . . such that

¯ ≤ dk x − x ¯ Ik (x) − Ik (x) (H6) There exists a constant 0 < L
0 such that ¯ s, ¯ φ¯) ≤ L∗ (|s − s| ¯ + φ − φ¯Bh ) for all s, s¯∈ J A(s)g(s, φ ) − A(s)g(

and

φ , φ¯ ∈ Bh .

For every n ∈ N, let us take here l¯n (t) = M1 Kn [L∗ + ln (t)] for the family of semi-norm { · n }n∈N . In what follows we fix τ > 1 and assume [M0 L∗ Kn +

m 1 + M1 ∑ dk ] < 1. τ k=1

Remark 1. It seems that conditions (H6) and (H8) are strong conditions but one can construct weaker conditions on an operator A(t) for boundedness and Lipschitz’s continuity on the phase space for the computation purpose. But it is a seperate article to study. The weaker conditions may be generated through monotone operator theory, refer [35] . Theorem 3. Suppose that hypotheses (H1)-(H8) are satisfied. Moreover ˆ n ˆ +∞ M1 Kn ds > max(L, p(s))ds for each s + ψ (s) 1 − M0 LKn 0 δn

n ∈ N.

with

δn = (Kn M1 H + Mn )φ Bh +

Kn [(M1 + 1)M0 L + M1Ln 1 − M0 LKn m

+M0 L[M1 (Kn H + 1) + Mn ]φ Bh + M1 ∑ ck ]. k=1

Then the impulsive neutral evolution problem (1) − (3) has a mild solution.

(5)

26

Dimplekumar N. Chalishajar, K. Karthikeyan, A. Anguraj / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 19–34

Proof. We transform problem (1) − (3) into a fixed point problem. Consider an operator N : B∗ → B∗ defined by ⎧ φ (t) if t ≤ 0, ⎪ ˆ t ⎪ ⎪ ⎪ ⎨ U (t, 0)[φ (0) − g(0, φ )] + g(t, y ) + U (t, s)A(s)g(s, y )ds t s N(y) = h ∈ B∗ : h(t) = 0 ˆ t ⎪ ⎪ ⎪ ⎪ + U (t, s) f (s)ds + ∑ U (t,tk )Ik (y(tk− )), t ∈ J, ⎩ 0

0 0 is a constant; (ii) there exist constants wk and C such that 0 ≤ wk ≤ C < 1, k = 1, 2, . . . such that ||Ik (x)|| ≤ wk ||x||

for x ∈ Rn .

 where ||x|| = ∑nk=1 x2k , x = (x1 , x2 , . . . , xn ). Then the trivial solution of the RIFrDE (10) is p-moment exponentially stable. Proof. The proof follows from Theorem 6.2 applied to V (x) = ||x||2 = xT x for x ∈ Rn and Lemma 1. Example 4 (Exponential stability of IFrDE with random moments of impulses). Let τi , i = 1, 2, . . . be independent exponentially distributed random variables with a parameter λ , i.e. E(τi ) = λ1 , i = 1, 2, . . . . Consider the initial value problem for the system of impulsive Caputo fractional differential equations with random moments of impulses c q 0 D x(t) c q 0 D y(t)

= − a(t)(x + y), = a(t)(x − y) for t ≥ 0, ξk < t < ξk+1 ,

x(ξk + 0) = Ax(ξk − 0), x(0) = x0 ,

y(ξk + 0) = By(ξk − 0) for k = 1, 2, . . . ,

y(0) = y0 .

where a ∈ C(R+ , R+ ) : a(t) ≥ m > 0 for t ≥ 0, m, A, B ∈ R : |A| < 1, |B| < 1 are constants.

(35)

62

to

Ravi Agarwal, Snezhana Hristova, Donal O’Regan / Discontinuity, Nonlinearity, and Complexity 6(1) 49–63

Consider the Lyapunov function V (x, y) = x2 + y2 = (x, y)T (x, y). Then condition 2(ii) in Corollary 3 reduces A2 x2 + B2 y2 ≤ C(x2 + y2 ).

where C = max{A2 , B2 }. Let (x(t), y(t)) ∈ Cq (R+ , R2 ) be a solution of (35). Condition 2(i) in Corollary 3 reduces to x(t)(−a(t)(x(t) + y(t))) + y(t)(a(t)(x(t) − y2 )) = −a(t)(x2 (t) + y2 (t)) ≤ −mV (x(t), y(t)), t ∈ R+ .

(36)

According to Corollary 3 the solution of RIFrDE (35) is exponentially stable in mean square. Now, consider the system without any impulses, i.e. the system of fractional differential equations c q 0 D x(t) c q 0 D y(t)

= − a(t)(x + y)

(37)

= a(t)(x − y) for t ≥ 0.

According to Remark 3 [21] and Corollary 1 [21] the zero solution of (37) is stable and according to [30] the inequality (x2 (t) + y2 (t))2 = V (x(t), y(t)) ≤ V (x0 , y0 ) = (x20 + y20 )2 holds. The presence of impulses at random time changes the behavior of the solution of the fractional differential equation. 2 2m(t)g(t) + m(t0 ) C q t0 D (m(t))

1 (t − t0

)q Γ(1 − q)

≤ (c − 2)m(t) + m(t0 )

+Ct0 Dq (m(t)) ≤ Cm(t), 1 (t − t0

)q Γ(1 − q)

= 0.

References [1] Agarwal R. and Hristova, S. (2012), Strict stability in terms of two measures for impulsive differential equations with supremum, Appl. Anal., 91, 1379-1392. [2] Hristova S. (2010), Integral stability in terms of two measures for impulsive functional differential equations, Math. Comput. Modell., 51, 100-108. [3] Hristova S. (2010), Stability on a cone in terms of two measures for impulsive differential equations with supremum , Appl. Math. Lett., 23, 5, 508-511. [4] Hristova S. (2009), Razumikhin method and cone valued Lyapunov functions for impulsive differential equations with supremum, Intern. J. Dynam. Syst. Diff. Eq., 2, 3-4, 223-236. [5] Hristova S., Stefanova K. (2012), Practical stability of impulsive differential equations with supremum by integral inequalities, Eur. J. Pure Appl. Math., 5, 1, 30-44. [6] Hristova S. (2010), Lipschitz stability for impulsive differential equations with supremum, Intern. Electr. J. Pure Appl. Math., 1, 4, 345-358. [7] Hristova S. (2009), Qualitative Investigations and Approximate Methods for Impulsive Differential Equations, Nova Sci. Publ. [8] Lakshmikantham V., Bainov D.D. , Simeonov P.S. (1989), Theory of Impulsive Differential Equations, World Scientific, Singapore. [9] Sanz-Serna J.M., Stuart A.M. (1999), Ergodicity of dissipative differential equations subject to random impulses, J. Diff. Equ., 155, 262-284. [10] Wu S., Hang D , Meng X. (2004), p-Moment Stability of Stochastic Equations with Jumps, Appl. Math. Comput., 152, 505-519. [11] Bagley, R.L. and Calico, R.A. (1991), Fractional order state equations for the control of viscoelasticallydamped structures, J. Guid., Contr. Dyn., 14(2), 304-311. [12] Laskin N. (2000), Fractional market dynamics, Phys. A, Stat. Mech. Appl., 287, 3-4, 482-492. [13] Anguraj, A., and Vinodkumar, A. (2010), Existence, uniqueness and stability results of random impulsive semi-linear differential systems, Nonlinear Anal. Hybrid Syst., 3, 475-483.

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[14] Anguraj, A., Ranjini, M.C., Rivero, M., and Trujillo, J. J. (2015), Existence results for fractional neutral functional differential equations with random impulses, Mathematics, 2015(3), 16-28. [15] Wang J.R., Feckan M., Zhou Y. (2016), Random Noninstantaneous Impulsive Models for Studying Periodic Evolution Processes in Pharmacotherapy, Mathematical Modeling and Applications in Nonlinear Dynamics, 14, Nonlinear Systems and Complexity, 87-107. [16] Lakshmikantham V., Leela S., Devi J.V. (2009), Theory of Fractional Dynamical Systems, Cambridge Scientific Publishers. [17] Podlubny I. (1999), Fractional Differential Equations, Academic Press, San Diego. [18] Das Sh. ( 2011), Functional Fractional Calculus, Springer-Verlag Berlin Heidelberg. [19] Diethelm K. (2010), The Analysis of Fractional Differential Equations, Springer-Verlag Berlin Heidelberg. [20] Devi J. V., Mc Rae F.A., Drici Z. (2010), Variational Lyapunov method for fractional differential equations, Comput. Math. Appl. 64, 2982-2989. [21] Aguila-Camacho, N., Duarte-Mermoud, M. A., and Gallegos, J. A. (2014), Lyapunov functions for fractional order systems, Comm. Nonlinear Sci. Numer. Simul., 19, 2951-2957. [22] Baleanu D., Mustafa O.G. (2010), On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl. 59, 1835-1841. [23] Agarwal, R., Benchohra, M., and Slimani, B. A. (2008), Existence results for differential equations with fractional order and impulses, Mem. Differ. Equ. Math. Phys, 44, 1-21. [24] Ahmad, B. and Sivasundaram, S. (2009), Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal. Hybrid Syst. 3, 251-258. [25] Benchohra M., Slimani B. A. (2009), Existence and uniqueness of solutions to impulsive fractional differential equations, Electronic Journal of Differential Equations, No. 10, 1-11. [26] Wang G., Ahmad B. , Zhang L., Nieto J. (2014), Comments on the concept of existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat. 19, 401-403. [27] Feckan M., Zhou Y., Wang J. (2012), On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17, 3050-3060. [28] Devi J. V., Mc Rae F.A., Drici Z. (2010), Generalized quasilinearization for fractional differential equations, Comput. Math. Appl. 59, 1057-1062. [29] Duarte-Mermoud M. A., Aguila-Camacho N., Gallegos J. A., Castro-Linares R. (2015), Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Comm. Nonlinear Sci. Numer. Simul., 22, 650-659. [30] Agarwal, R., Hristova, S., and O’Regan, D. (2015), Lyapunov functions and strict stability of Caputo fractional differential equations, Adv. Diff. Eq., 2015. [31] Agarwal R., O’Regan D., Hristova S. (2015), Stability of Caputo fractional differential equations by Lyapunov functions, Appl. Math., 60, 6, 653-676. [32] Agarwal, R., Hristova S., and O’Regan, D. (2016), Practical stability of Caputo fractional differential equations by Lyapunov functions, Diff. Eq. Appl. 8, 1, 53-68.

Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 65–74

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

An Analytic Technique for the Solutions of Nonlinear Oscillators with Damping Using the Abel Equation A Ghose-Choudhury1†, Partha Guha2† 1 2

Department of Physics, Surendranath College, 24/2 Mahatma Gandhi Road, Calcutta 700009, India SN Bose National Centre for Basic Sciences JD Block, Sector III, Salt Lake Kolkata 700098, India Submission Info Communicated by Valentin Afraimovich Received 24 May 2016 Accepted 16 June 2016 Available online 1 April 2017 Keywords Li´enard equation Abel equation Chiellini integrabilty condition

Abstract Using the Chiellini condition for integrability we derive explicit solutions for a generalized system of Riccati equations x¨ + α x2n+1 x˙ + x4n+3 = 0 by reduction to the first-order Abel equation assuming the parameter α ≥  2 2(n + 1). The technique, which was proposed by Harko et al, involves use of an auxiliary system of first-order differential equations sharing a common solution with the Abel equation. In the process analytical proofs of some of the conjectures made earlier on the basis of numerical investigations in [1] is provided.

©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Second-order ordinary differential equations (ODEs) with linear damping are the most commonly studied extensions of undamped motion, the simplest example being the case of damped oscillations x¨ + γ x˙ + ω 2 x = 0 which admits a closed-form solution. In the case of nonlinear ODEs even with linear damping the construction of a closed form solution is often a nontrivial task and such equations often display a variety of interesting phenomena such as chaos in the case of non-autonomous nonlinear terms, complex periodicity, limit cycles etc. An equation of the form x˙ + f (x)x˙ + g(x) = 0 where f (x) and g(x) are arbitrary C ∞ (I) real-valued functions of x defined on a real interval I ⊆ R is known as a Li´enard equation [2]. There exists a vast literature on this equation alone as it is the favored equation for modelling several phenomena ranging from electrical circuits, heart beat activity, neuron activity, chemical kinetics to turbulence in fluid dynamics [3–6]. Mathematical techniques such as those of Lie symmetries [7, 8] and Wierstrass integrability have been used to analyse the Li´enard equation [9]. Its generalization the Levinson-Smith equation x¨ + f (x, x) ˙ x˙ + g(x) = 0 [10] has found applications in astrophysics where for instance the time dependence of perturbations of the stationary solutions of spherically symmetric accretion processes is modelled by an equation of this form [11]. † Corresponding

author. Email address: [email protected], [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2017.03.006

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From a practical point of view the assumption of linear damping is often insufficient. Generally it is possible to divide oscillator systems into two broad categories, those with linear damping and nonlinear elasticity and those with nonlinear damping and linear elasticity. Extensive studies of both these categories may be found in [12–15]. Various quantitative methods have been employed for their analysis depending on the context and convenience such as the method of multiple scales, successive approximation, averaging method besides qualitative studies [16–18]. The following generalization of the Li´enard equation involving a quadratic dependance on the velocity besides the usual linear damping term, viz x¨ + g2 (x)x˙2 + g1 (x)x˙ + g0 (x) = 0,

(1)

was studied by Bandic [19]. Special cases of this equation corresponding to g1 (x) = 0 naturally occur for oscillators involving a variable mass and are derivable from a Lagrangian of the form L(x, x) ˙ = 12 m(x)x˙2 + V (x). Recently Kovacic and Rand [20] studied several examples of a position-dependent coefficient of the kinetic energy, which stem from a position-dependent mass or are the consequence of geometric/kinematic constraints. Some notable examples of position-dependent mass systems include the Mathews-Lakshmanan oscillator equation [21], which has also been studied in the quantum regime, the quadratic Loud systems [22] and the Cherkas system [23]. In [24] Cveti´canin analysed the case of strong quadratic damping with a model ˙ x| ˙ = 0. The issue of isochronicity in equations of the Li´enard type has also been equation given by x¨ + x + 2δ x| extensively studied [25]. In [1] a variant of the generalized Riccati system of equations, viz x¨ + α x2n+1 x˙ + x4n+3 = 0,

(2)

was considered. It was established on the basis of numerical studies that for α much smaller than a critical value the dynamics is periodic, the originbeing a centre. Furthermore the solution changes from being periodic to aperiodic at a critical value αc = 2 2(n + 1), which is independent of the initial conditions. This behaviour is explained by finding a scaling argument via which the phase trajectories corresponding to different initial conditions collapse onto a single universal orbit. Numerical evidence for the transition is shown. Further, using a perturbative renormalization group argument, it is conjectured that the oscillator, x¨ + (2n + 3)x2n+1 x˙ + x4n+3 + w20 x = 0,

(3)

exhibits isochronous oscillations. The correctness of the conjecture is established numerically. In this communication we provide analytical proofs for some of these assertions. Equation (1) may be reduced to a first-order ODE by means of the transformation x˙ = 1/v, namely dv = g0 (x)v3 + g1 (x)v2 + g2 (x)v = F(x, v). dx

(4)

This is an Abel equation of the first-order and first kind and may be viewed as a generalization of the Riccati equation. Such equations, which first appeared in course of Abel’s investigations of the theory of elliptic functions, usually arise in problems involving the reduction of order of second and higher-order equations and are frequently encountered in modelling of practical problems, e.g., the Emden equation, the van der Pol equation etc. They are also relevant in the study of quadratic systems in the plane [26], the centre-focus problem [27] and in certain cosmological models [28]. 1.1

Derivation of the Chiellini condition for integrability

Recently Harko et al [29,30] have considered certain exactly integrable cases of the Li´enard equation by appealing to an integrability criterion known as the Chiellini condition and making use of the first-order Abel equation.

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67

´ ´ Multiplying (4) by exp(− g2 (x)dx) and setting u = v exp(− g2 (x)dx) leads us to the standard form of the Abel equation of the first kind namely [31] du = A(x)u2 + B(x)u3 , (5) dx ´ ´ where A(x) = g1 (x) exp( g2 (x)dx) and B(x) = g0 (x) exp(2 g2 (x)dx) respectively. An exact solution of (5) can be constructed provided the Chiellini condition for integrability [32, 33] for (5), given by d B ( ) = sA(x), (6) dx A is satisfied where s is a nonzero constant [29]. When g1 (x) = 0, the above condition becomes g0 g2 d g0 ( ) = sg1 − . (7) dx g1 g1 In order to derive this condition let us consider the following generalized version of the Li´enard equation, viz

x¨ + gn (x)x˙n + g0 (x) = 0.

(8)

Set x˙ = ξ (x), so that (8) becomes dξ . (9) dx Suppose ξ = F(x)G(u(x)), where G is a function of u. By differentiating ξ with respect to x and substituting it back into (9) we obtain FF  G2 + gn (x)F n Gn + g0 (x) . (10) u = − F 2 G ∂∂Gu In order to separate the variables and integrate equation (10), we observe that the function F should satisfy:

ξ ξ  + gn (x)ξ n + g0 (x) = 0,

ξ =

g0 (x) F = kgn (x)F n−2 = l 2 , F F where k and l and constants or in other words F kgn (x) = n−1 , lg0 (x) = FF  . F From these relations we obtain ˆ l g0 = F n. F 2−n = (2 − n) kgn (x)dx and k gn whence we have ˆ k2 d g0 ( ) = (2 − n)gn (x)((2 − n)k gn (x)dx)2(n−1)/(2−n) . dx gn l Now suppose G = u, then (10) reduces to k k u = −kgn (x)F n−2 F n−2 u − gn (x)F n−2 un−1 − F n−2 gn (x) = −gn (x)F n−2 (ku + un−1 + ). lu lu This being separable it is solvable. Setting n = 1, (11) reduces to k2 d g0 ( ) = g1 (x), dx g1 l while from (1) it follows that when g2 = 0, dv = g1 v2 + g0 v3 . dx which is to be compared with (5). It is now obvious that the criterion stated in (6) is identical to (13) s = k2 /l.

(11)

(12)

(13)

with

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1.2

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Construction of an implicit solution

As explained in [29] an implicit solution of (5) can be accomplished by defining a new variable w = uB/A and using the Chiellini condition such that (5) is transformed to dw A2 = w(w2 + w + s). dx B

(14)

This leads to a separation of the variables, namely ˆ F(w, s) :=

dw = w(w2 + w + s)

ˆ

A2 1 dx = B s

ˆ d ln(B/A).

(15)

where the Chiellini condition has been used once again and finally allows us to express the solution of (14) in the implicit form B (16) | | = K −1 esF(w,s) . A where K −1 is an arbitrary constant of integration. It follows that x˙ =

g0 (x) 1 B 1 = = ´ . = ´ g dx 2 v ue g2 dx g (x)w(x) Ae w(x) 1

and hence

ˆ t − t0 =

w(x)g1 (x) dx. g0 (x)

(17)

The form of the right-hand-side of (16) depends on the value of the parameter s and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

w 1 2w + 1 √ exp(− √ arctan( √ )) 4s − 1 4s − 1 w2 + w + s 1 w ) exp( esF(w,s) = 1 2w + 1 ⎪ w+ 2 ⎪ ⎪ ⎪ 1 1 ⎪ 1 + 2w − 2√1−4s 1 + 2w 2√1−4s w ⎪ ⎩√ |1 − √ | | |1 + √ 1 − 4s 1 − 4s w2 + w + s

1 s> , 4 1 s= , 4 1 s< . 4

(18)

2 Solution of first-order ODEs via an auxiliary system of ODEs In [1] the behaviour of the dynamical system described by the equation x¨ + α x2n+1 x˙ + x4n+3 = 0,

(19)

the system was analysed and it was conjectured that there exists a critical value of the parameter αc below which admits closed orbits. Extensive numerical computations indicated that the critical value was αc = 2 2(n + 1). In view of the method described above one can obtain analytically this critical value by reducing the equation to a first-order Abel equation: dv = α x2n+1 v2 + x4n+3 v3 . (20) dx It is observed that the Cheillini integrability condition is satisfied with the constant s = 2(n + 1)/α 2 . From (18) it is seen that the nature of the solution for w changes as s varies from less  than 1/4 to greater than 1/4. The critical value corresponding to s = 1/4 implies that the parameter αc = 2 2(n + 1). This provides a proof of the validity of the conjecture made in [1].

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A useful method of solving a first-order ordinary differential equation (FOODE) is by the introduction of an auxiliary system of first-order ODEs which have a common solution with the given equation [29,30]. To explain how this is achieved consider a first-order ODE given by dv = F(x, v), dx

(21)

and introduce an auxiliary system of first-order ODEs dv = −F1 (x, v) + G(x) f (v), dx 1 dv 1 = F2 (x, v) + G(x) f (v), dx 2 2

(22) (23)

subject to the constraint F1 + F2 = F, where G(x) is a function to be determined. If a function G(x) exists such that (22) and (23) have a common solution then it is easy to show that this solution satisfies the equation (21). The above technique can be adapted to deal with second-order ODEs which frequently arise in physical applications. Consider a second-order ODE of the form (1), viz dx dx d2x + g1 (x) + g2 (x)( )2 + g0 (x) = 0. 2 dt dt dt

(24)

Typically if g2 = 0 then we have an equation of the Li´enard type, and if g1 = 0 we obtain an equation with a quadratic dependance on the velocity which, from a Newtonian point of view, may be interpreted as arising from the dependance of the mass of a particle on its position coordinate. Both types of equations having either a linear or a quadratic dependance on the velocity have been extensively studied [23, 25, 34, 35]. The transformation dx/dt = 1/v(x) causes (24) to become dv = g0 v3 + g1 v2 + g2 v := F(x, v). dx

(25)

Demanding F1 = F(x, v), F2 = 0 and f (v) = v3 the analogs of (22) and (23) then have the following forms, in terms of the transformed variables, namely: dv = (G(x) − g0 )v3 − g1 v2 − g2 v, dx dv 1 = G(x)v3 . dx 2 The use of the Chiellini condition for (26) allows us to express G(x) as ˆ ˆ ˆ G(x) = g0 + g1 exp( g2 dx)[Γ + s g1 exp(− g2 dx)dx],

(26) (27)

(28)

where Γ is a constant of integration with the constant s appearing as a result of the use of the Chiellini condition. Notice that owing to the convenient choices made for the functions F1 and F2 , (27) is separable and its solution is given by  ˆ 1 dx = = ± B − G(x)dy, (29) v dt with B being a constant of integration. Now the existence of a common solution means that dv 1 = G(x)v3 = (G(x) − g0 )v3 − g1 v2 − g2 v, dx 2

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which implies upon using (29)  ˆ ˆ 2g2 G(x) − 2g0 = ±2 B − G(x)dx + (B − G(x)dx). g1 g1

(30)

Eqn (30) may be used to determine the values of the parameters s , Γ and B after substituting the value of G(x) from (28). Knowledge of G(x) then allows us to obtain the common solution from (29) in the form ˆ dx , (31) ±t − t0 =  ´ B − G(x)dx with t0 being a parameter which defines the families of solutions. The procedure is illustrated below. Example 1. x¨ + α x2n+1 x˙ + x4n+3 + w20 x2n+1 = 0 Under the transformation x˙ = 1/v this equation becomes dv = (α x2n+1 )v2 + (x4n+3 + w20 x2n+1 )v3 := F(x, v). dx Choose the auxiliary system of FOODEs to be the following: dv = −F(x, v) + G(x)v3 = (G(x) − x4n+3 − w20 x2n+1 )v3 − α x2n+1 v2 , dx dv 1 = G(x)v3 . dx 2 Applying the Cheillini integrability condition to the first of these equations we have d G(x) − x4n+3 − w20 x2n+1 ( ) = s (−α x2n+1 ). dx −α x2n+1 We solve this for G(x) to get G(x) = x4n+3 (

s α 2 + 1) + (α Γ + w20)x2n+1 . 2(n + 1)

Upon solving the second auxiliary FOODE, which is separable, we obtain  ˆ 1 = ± B − G(x)dx v

(32)

(33)

where B and Γ are arbitrary constants of integration. If a common solution exists for the two auxiliary FOODEs then we must have 1 G(x)v3 = (G(x) − x4n+3 − w20 x2n+1 )v3 + (−α x2n+1 )v2 . 2 which leads  ˆ 2n+1 2(n+1) 2 [(x + w0 ) ± α B − G(x)dx]. (34) G(x) = 2x Equating (32) and (34) we have upon equating coefficients of different powers of x (with ξ = α 2 s /2(n + 1) − 1), α2 (ξ + 2). ξ2 = − (35) n+1

A Ghose-Choudhury, Partha Guha / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 65–74

ξ (α Γ − w20 ) = −

α2 (α Γ + w20 ), (α Γ − w20 )2 = 4α 2 B. n+1

We can solve for the constants of integration and ξ to obtain  1 α2 α2 2 α2 )± ( ) − 8( )]. ξ = [−( 2 n+1 n+1 n+1 Γ=

α 2 w40 w20 (n + 1)ξ − α 2 [ ], B = . α (n + 1)ξ + α 2 [(n + 1)ξ + α 2 ]2

71

(36)

(37)

(38)

Knowing the constants of integration, the solution may be reduced to quadrature using (32) and (33), i.e., ˆ dx . (39) ±t − t0 = w20 ξ ξ +2 4(n+1) 2(n+1) B − 4(n+1) x − [(n+1)ξ +α 2 ] x Case A: α = 2n + 3 and w0 = 0 For this choice of the parameter α we find from (37) since s = 2(n + 1)(ξ + 1)/α 2 (s+ , s− ) = (−

2(n + 2) 2(n + 1)(4n + 5) ,− ), (2n + 3)2 (2n + 3)2

(B+ , B− ) = ( (Γ+ , Γ− ) = (−

w40 , w4 ), 4(n + 1)2 0

(n + 2) (4n + 5) 2 w20 , − w ). (n + 1)(2n + 3) (2n + 3) 0

These values lead to the following expressions for the unknown function G(x), viz, G+ (x) = −

1 [x4n+3 + w20 x2n+1 ], n+1

G− (x) = −4(n + 1)[x4n+3 + w20 x2n+1 ]. which in turn yield the solutions ±t − t0+

ˆ = 2(n + 1)

±t − t0− =

ˆ

dx x2(n+1) + w20

,

dx . x2(n+1) + w20

respectively. It is evident from these solutions that they are equivalent up to a scaling. Indeed setting w0 = 1 one may explicitly express the solution in terms of the hypergeometric function 2 F1 (a, b; c; x) because ˆ 1 1 dx = x2 F1 (1, ;1+ ; −x2(n+1) ). 2(n + 1) 2(n + 1) x2(n+1) + 1 From (37) it is evident thatα 2 = (2n + 3)2 > 8(n + 1). The critical value of α corresponding to the vanishing of the discriminant is αc = 2 2(n + 1). Thus when α > αc and n = 0 we obtain the case of periodic motion. Incidently this corresponds to isochronous motion, in which the period function is independent of the initial condition. This is easily verified from the corresponding criterion given by Sabatini in [25]. There it is shown that

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Fig. 1 Graph for n = 0 − 10, for n = 0 we obtain arctan and for large n curves are dense

. for a Li´enard equation x+ ¨ f (x)x+g(x) ˙ = 0 having an isochronous center at the origin with f , g ∈ C1 (J, R), f (0) = g(0) = 0, g (0) > 0 the forcing term g(x) must be of the form ˆ x 1 g(x) = g (0)x + 3 ( s f (s)ds)2 . x 0 It is straightforward to verify that these conditions are satisfied by the equation x¨ + (2n + 3)x2n+1 x˙ + x4n+3 + w20 x = 0, and hence by the equation x¨ + 3xx˙ + x3 + w20 x = 0, which corresponds to n = 0. Case B: α = 2n + 3 and w0 = 0 When w0 = 0 we have from (38) that Γ = B = 0 and the solutions are x=[

1 1 1 2(n + 1) ]1/(2n+1) and x = [ ]1/(2n+1) . 2n + 1 (t0+ ∓ t) 2n + 1 (t0− ∓ t)

respectively and are singular.

Fig. 2 Graph for n = 1 − 10, where t0− and t0+ are approaching from lhs and rhs of t = 0

.

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73

3 Generalizations of the Chiellini condition The Chiellini integrability condition has been used in a number of works (see [30] and references therein). Its generalization to the case when higher powers of u appear in the right hand side of (5) has also been studied. In view of it efficacy in deriving solutions of the first-order Abel equation we consider below higher-order generalizations of the Li´enard equation. 3.1

Higher-order Li´enard equation

Consider the higher-order Li´enard equation x¨ + f (x)x˙n+1 + g(x)x˙n = 0.

(40)

Suppose x˙ = ξ (x), so that (40) becomes

ξ  + f (x)ξ n + g(x)ξ n−1 = 0.

(41)

Once again we assume ξ = F(x)G(u(x)), where G is a function of u. Following the procedure outlined in Section 4.1 we obtain F  F n−3 Gn + f (x)F 2n−3 G2n−1 + g(x) . (42) u = − F n−2 Gn−1 ∂∂Gu After separating the variables we have g(x) F = k f (x)F n−1 = l n−2 , F F and this leads to the generalized Chiellini condition [30] l n−1 gn gn g ( ) = n−2 ( n−1 ) ≡ K( n−1 ), f k f f

(43)

which for n = 0 reduces to the usual Chiellini condition stated in (13). Upon introducing the transformation

ξ =( (41) becomes

g(x) )η (x), f (x)

d η (x) gn−1 (x) n = n−2 (η + η n−1 + K η ). dx f (x)

(44)

which is clearly separable. Acknowledgement The authors wish to thank Professors J. K Bhattacharjee and A. Mallik for their interest and encouragement. One of us (PG) wishes to acknowledge Professor Tudor Ratiu for his gracious hospitality at the Bernoulli Centre, EPFL during the fall semester of 2014, where part of this work was done. References [1] Sarkar, A., Partha Guha, A., Ghose-Choudhury, Bhattacharjee, J. K., Mallik, A. K., and Leach, P. G. L. (2012), On the properties of a variant of the Riccati system of equations, J. Phys. A: Math. Theor., 45, 415101 (9pp).

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[2] Li´enard, A. (1928), Revue g´en´erale de l’´electricit´e, 23, 901– 912, and 946–954. [3] van der Pol, B. (1927), On relaxtion-oscillations, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 2, 978–992. [4] Van der Pol, B. and van der Mark, J. (1928), The heart beat considered as a relaxation oscillations and an electrical model of the heart, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 6, 763–775. [5] Fitzhugh, F. (1928), Impulses and physiological states in theoretical models of nerve membranes. Biophysics Journal, 1, 445-466. [6] Strogatz, S. H. (1994), Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, Massachussets. [7] Garcia, I. A., Gin´e, J., and Llibre, J. (2008), Li´enard and Riccati differential equations related vis Lie algebras. Discrete Continuous Dynamical Systems B, 10, 485–494. [8] Carinena, J. F. and de Lucas, J. (2011), Lie systems: theory, generalizations, and applications. Dissertationes Mathematicae (Rozprawy Matematyczne), 479, 1–162. [9] Gin´e, J. and Llibre, J. (2010), Weierstrass integrability of differential equations. Applied Mathematics Letters, 23, 523-526. [10] Levinson, N. and Smith, O. (1942), A general equation for relaxation oscillations, Duke Mathematical Journal, 9, 382–403. [11] Ran, Z. (2009), One exactly soluble model in isotropic turbulence. Advances and Applications in Fluid Mechanics, 5, 41–47. [12] Mickens, R. E. (2002), Analysis of non-linear oscillators having non-polynomial elastic terms, J. Sound Vib, 255, 789–792. [13] Waluya, S. B. and van Horssen, W. T. (2003), On the periodic solutions of a generalized non-linear Van der Pol oscillator, J. Sound Vib, 268, 209–215. [14] Pilipchuk, V. N. (2007), Strongly nonlinear vibrations of damped oscillators with two nonsmooth limits, J. Sound Vib, 302, 398-402. [15] Nayfeh, A. H. and Mook, D. (1979), Nonlinear Oscillations, Wiley, New York. [16] Bogolyubov, N. N. and Mitropolskii, J. A. (1974), Asimptoticheskie metodi v teorii nelinejnih kolebanij, Nauka Moskva. [17] Magnus, K. (1997), Schwingungen, Teubner, Stuttgart. [18] Andronov, A. A., Vitt, A. A., and Hajkin, S. E. (1981), Teorija kolebanij, Nauka, Moskva. [19] Bandic, I. (1961), Sur le crit`ere d’int´egrabilit´e de l’´equation diff´erentielle g´en´eralis´ee de Li´enard, Bollettino dell Unione Matematica Italiana, 16, 59–67. [20] Kovacic, I. and Rand, R. (2013), About a class of nonlinear oscillators with amplitude-independent frequency, Nonlinear Dynam., 74(1–2), 455–465. [21] Mathews, P. M. and Lakshmanan, M. (1974), On a unique nonlinear oscillator, Quart. Appl. Math. 32, 215. [22] Loud, W.S. (1964), The behavior of the period of solutions of certain plane autonomous systems near centers, Contr. Differential Equations, 3, 21–36. [23] Ghose Choudhury, A. and Guha, P. (2010), On isochronous cases of the Cherkas system and Jacobi’s last multiplier, J. Phys. A: Math. Theor., 43, 125202. [24] Cveticanin, L. (2009), Oscillator with strong quadratic damping force, Publ. Inst. Math. (Beograd) (N.S.), 85(99), 119–130. [25] Sabatini, M. (1999), On the period Function of Li´enard Systems, J. Diff. Eqns., 152, 467–487. [26] Alvarez, M. J., Gasull, A., and Giacomini, H. (2007), A new uniqueness criterion for the number of periodic orbits of Abel equation, J. Diff. Eqn., 234, 161–176. [27] Briskin, M., Francoise, J. P., and Yomdin, Y. (1998), The Bautin ideal of the Abel equation, Nonlinearity, 11, 431–443. [28] Yurov, A. V. and Yurov, V. A. (2008), Friedmann versus Abel equations: A connection unraveled, arXiv: 0809.1216v2. [29] Harko, T., Lobo, F. S. N., and Mak, M. K. A class of exact solutions of the Li´enard type ordinary non-linear differential equation arXiv:1302.0836v3[math-ph]. [30] Harko, T., Lobo, F. S. N., and Mak, M. K. (2013), A Chiellini type integrability condition for the generalized first kind Abel differential equation, Universal Journal of Applied Mathematics, 1, 101–104. [31] Cari˜nena, J. F., Ra˜nada, M. F., and Santander, M. (2004), One-dimensional model of a quantum nonlinear harmonic oscillator, Rep. Math. Phys., 54, 285. [32] A. Chiellini, Sull’integrazione dell’equazione differenziale y + Py2 + Qy3 = 0, Bollettino dell’Unione Matematica Italiana, 10, 301-307 (1931). [33] Kamke, E. (1971), Differentialgleichungen, Losungsmethoden und Losungen, Nauka, Moskva. [34] Guha P. and Ghose Choudhury, A. (2013), The Jacobi last multiplier and isochronicity of Li´enard type systems, Rev. Math. Phys., 25(6), 1330009. [35] Raouf Chouikha, A. (2007), Isochronous centers of Lienard type equations and applications, J. Math. Anal. Appl., 331, 358376.

Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 75–86

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

On the Existence of Stationary Solutions for Some Systems of Non-Fredholm Integro-Differential Equations with Superdiffusion Vitali Vougalter1†, Vitaly Volpert2,† 1 2

Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, 69622, France Submission Info Communicated by Dmitri Volchenkov Received 19 June 2016 Accepted 20 June 2016 Available online 1 April 2017

Abstract We establish the existence of stationary solutions for certain systems of reaction-diffusion equations with superdiffusion. The corresponding elliptic problem involves the operators with or without Fredholm property. The fixed point technique in appropriate H 2 spaces of vector functions is employed.

Keywords Solvability conditions Non Fredholm operators Systems of integro-differential equations Stationary solutions Superdiffusion

©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Let us recall that a linear operator L acting from a Banach space E into another Banach space F has the Fredholm property when its image is closed, the dimension of its kernel and the codimension of its image are finite. As a consequence, the equation Lu = f is solvable if and only if φk ( f ) = 0 for a finite number of functionals φk from the dual space F ∗ . These properties of Fredholm operators are broadly used in various methods of linear and nonlinear analysis. Elliptic equations studied in bounded domains with a sufficiently smooth boundary satisfy the Fredholm property when the ellipticity condition, proper ellipticity and Lopatinskii conditions are satisfied (see e.g. [1–3]), which is the main result of the theory of linear elliptic equations. When working in unbounded domains, these conditions may not be sufficient and the Fredholm property may not be fulfilled. For example, for the Laplace operator, Lu = Δu considered in Rd Fredholm property does not hold when the problem is studied either in H¨older spaces, such that L : C2+α (Rd ) → Cα (Rd ) or in Sobolev spaces, L : H 2 (Rd ) → L2 (Rd ). For linear elliptic equations considered in unbounded domains the Fredholm property is satisfied if and only if, in addition to the conditions stated above, the limiting operators are invertible (see [4]). In some trivial cases, † Corresponding

author. Email address: [email protected], [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2017.03.007

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the limiting operators can be constructed explicitly. For example, when Lu = a(x)u + b(x)u + c(x)u,

x ∈ R,

with the coefficients of the operator having limits at infinity, a± = limx→±∞ a(x),

b± = limx→±∞ b(x),

c± = limx→±∞ c(x),

the limiting operators are given by L± u = a± u + b± u + c± u. Because the coefficients here are constants, the essential spectrum of the operator, which is the set of complex numbers λ for which the operator L − λ does not have the Fredholm property, can be found explicitly via the standard Fourier transform, such that

λ± (ξ ) = −a± ξ 2 + b± iξ + c± ,

ξ ∈ R.

The limiting operators are invertible if and only if the origin does not belong to the essential spectrum. For general elliptic equations the analogous assertions hold. The Fredholm property is satisfied when the essential spectrum does not contain the origin or when the limiting operators are invertible. These conditions may not be written explicitly. For non-Fredholm operators we may not apply the standard solvability conditions and in a general case solvability conditions are unknown. However, solvability conditions were obtained recently for some classes of operators. For example, consider the following problem Lu ≡ Δu + au = f .

(1)

in Rd , d ∈ N with a positive constant a. Here the operator L and its limiting operators coincide. The corresponding homogeneous problem has a nontrivial bounded solution, such that the Fredholm property is not satisfied. Because the differential operator contained in (1) has constant coefficients, we are able to obtain the solution explicitly by applying the standard Fourier transform. In [5] we derived the following solvability relations. Let f (x) ∈ L2 (Rd ) and x f (x) ∈ L1 (Rd ). Then equation (1) has a unique solution in H 2 (Rd ) if and only if ( f (x),

eipx d

(2π ) 2

)L2 (Rd ) = 0,

d p ∈ S√ a

a.e.

Here and below Srd denotes the sphere in Rd of radius r centered at the origin. Thus, although the Fredholm property is not satisfied for this operator, we can formulate solvability relations similarly. Note that this similarity is only formal because the range of the operator is not closed. In the situation when the operator contains a scalar potential, such that Lu ≡ Δu + b(x)u = f , we are not able to use the standard Fourier transform directly. However, solvability relations in three dimensions can be obtained by virtue of the spectral and the scattering theory of Schr¨odinger type operators (see [6]). Analogously to the constant coefficient case, solvability relations are expressed in terms of orthogonality to solutions of the adjoint homogeneous equation. We derive solvability conditions for several other examples of non Fredholm linear elliptic operators (see [4–13]). Solvability relations are crucial in the analysis of nonlinear elliptic problems. In the presence of nonFredholm operators, in spite of some progress in studies of linear equations, nonlinear non-Fredholm operators were analyzed only in few examples (see [5, 14–17]). Evidently, this situation can be explained by the fact that

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77

the majority of methods of linear and nonlinear analysis rely on the Fredholm property. In the present article we study some systems of N nonlinear integro-differential reaction-diffusion type equations, for which the Fredholm property may not be satisfied: √ ∂ uk = − −Δuk + ∂t

ˆ Ω

Gk (x − y)Fk (u1 (y,t), u2 (y,t), ..., uN (y,t), y)dy + ak uk , 1 ≤ k ≤ N.

(2)

N d Here {ak }√ k=1 are nonnegative constants, Ω ⊆ R , d = 1, 2, 3 are the more physically relevant dimensions. The operator −Δ is defined via the spectral calculus. System (2) describes a particular case of superdiffusion actively treated in the context of various applications in plasma physics and turbulence (see e.g. [18,19]), surface diffusion (see e.g. [20, 21]), semiconductors (see e.g. [22]) and so on. The superdiffusion can be understood as a random process of particle motion characterized by the probability density distribution of jump length. The moments of this density distribution are finite for normal diffusion, but this is not the case for superdiffusion. Asymptotic behavior at infinity of the probability density function determines the value of the power of the negative Laplacian (see e.g. [23]). In population dynamics the integro-differential equations are used to describe biological systems with intraspecific competition and nonlocal consumption of resources (see e.g. [24–26]). The stability issues for the travelling fronts of reaction- diffusion type problems with the essential spectrum of the linearized operator crossing the imaginary axis were also treated in [27, 28]. Note that the single equation of (2) type has been studied in [29]. Reaction-diffusion type problems in which in the diffusion term the Laplacian is replaced by the nonlocal operator with an integral kernel were treated in [30]. The nonlinear terms of system (2) will fulfill the following regularity requirements.

Assumption 1. Functions Fk (u, x) : RN × Ω → R, 1 ≤ k ≤ N are such that 

N

∑ Fk2(u, x) ≤ Q|u|R

N

+ h(x)

f or

u ∈ RN , x ∈ Ω,

(3)

k=1

with a constant Q > 0 and h(x) : Ω → R+ , h(x) ∈ L2 (Ω). Furthermore, they are Lipschitz continuous functions, such that  N

∑ (Fk (u(1) , x) − Fk (u(2) , x))2 ≤ l|u(1) − u(2) |R

N

f or

any

u(1),(2) ∈ RN ,

x ∈ Ω,

(4)

k=1

where a constant l > 0.

 Here and below we use the notations for a vector u := (u1 , u2 , ..., uN ) ∈ RN and its norm |u|RN := ∑Nk=1 u2k . Evidently, the stationary solutions of problem (2), if any exist, will satisfy the system of nonlocal elliptic equations ˆ √ − −Δuk + Gk (x − y)Fk (u1 (y), u2 (y), ..., uN (y), y)dy + ak uk = 0, ak ≥ 0, 1 ≤ k ≤ N. Ω

For the technical purposes we consider the auxiliary semi-linear system ˆ √ −Δuk − ak uk = Gk (x − y)Fk (v1 (y), v2 (y), ..., vN (y), y)dy, 1 ≤ k ≤ N. Ω

(5)

´ We denote ( f1 (x), f2 (x))L2 (Ω) := Ω f1 (x) f¯2 (x)dx, with a slight abuse of notations in the case when these functions do not belong to L2 (Ω), like for example those used in the orthogonality relations of the assumption below. Indeed, if f1 (x) ∈ L1 (Ω) and f2 (x) is bounded there, then the integral over Ω mentioned above is well

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defined. We begin the article with the treatment of the whole space case, such that Ω = Rd and the corresponding Sobolev space is equipped with the norm u 2H 2 (Rd , RN ) :=

N

N

∑ uk 2H (R ) = ∑ { uk 2L (R ) + Δuk 2L (R ) }, 2

d

k=1

2

d

2

d

k=1

√ where u(x) : Rd → RN . The primary obstacle in solving problem (5) is that operators −Δ − ak : H 2 (Rd ) → L2 (Rd ), ak ≥ 0 fail to satisfy the Fredholm property. The similar situations in linear equations, which can be selfadjoint or non self-adjoint involving non Fredholm second, fourth and sixth order differential operators or even systems of equations including non Fredholm operators have been treated actively in recent years (see [6,8–13]). We are able to prove that system of equations (5) defines a map Ta : H 2 (Rd , RN ) → H 2 (Rd , RN ), ak ≥ 0, 1 ≤ k ≤ N, which is a strict contraction under stated technical conditions. We make the following assumption on the integral kernels contained in the nonlocal parts of system (5). Assumption 2. Let Gk (x) : Rd → R, Gk (x) ∈ W 1,1 (Rd ), 1 ≤ k ≤ N, 1 ≤ d ≤ 3 and 1 ≤ m ≤ N − 1, m ∈ N with N ≥ 2. I) Let ak > 0, 1 ≤ k ≤ m, assume that xGk (x) ∈ L1 (Rd ) and e±iak x (Gk (x), √ )L2 (R) = 0 when d = 1. 2π (Gk (x),

eipx d

(2π ) 2

)L2 (Rd ) = 0 f or p ∈ Sadk a.e. when d = 2, 3.

(6)

(7)

II) Let ak = 0, m + 1 ≤ k ≤ N, assume that xGk (x) ∈ L1 (Rd ) and (Gk (x), 1)L2 (Rd ) = 0. Let us use the hat symbol here and below to designate the standard Fourier transform, such that ˆ 1  Gk (x)e−ipx dx, p ∈ Rd . Gk (p) := d (2π ) 2 Rd Thus k (p) L∞ (Rd ) ≤ G

1 d

(2π ) 2

(8)

(9)

Gk L1 (Rd ) .

Let us introduce the following auxiliary quantities k (p) k (p) G p2 G L∞ (Rd ) , ∞ d }, 1 ≤ k ≤ m. |p| − ak |p| − ak L (R )

(10)

k (p) G k (p) L∞ (Rd ) }, m + 1 ≤ k ≤ N. L∞ (Rd ) , pG p

(11)

Mk := max{

Mk := max{

Note that expressions (10) and (11) are finite by virtue of Lemma A1 in one dimension and Lemma A2 for d = 2, 3 of the Appendix of [29] under our Assumption 2. Therefore, we define M := maxMk , 1 ≤ k ≤ N. with Mk given by (10) and (11). We have the following proposition.

(12)

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79

√ d Theorem 1. Let Ω = Rd , d = 1, 2, 3, Assumptions 1 and 2 hold and 2(2π ) 2 Ml < 1. Then the map Ta v = u on H 2 (Rd , RN ) defined by the system of equations (5) possesses a unique fixed point va (x) : Rd → RN , which is the only stationary solution of problem (2) in H 2 (Rd , RN ). This fixed point va (x) is nontrivial provided the  intersection of supports of the Fourier transforms of functions suppF k (0, x)(p) ∩ suppGk (p) is a set of nonzero d Lebesgue measure in R for some 1 ≤ k ≤ N. Then we turn our attention to the studies of the analogous problem on the interval Ω = I := [0, 2π ] with periodic boundary conditions for the solution vector function and its first derivative. We assume the following about the integral kernels present in the nonlocal parts of system (5) in such case. Assumption 3. Let Gk (x) : I → R, Gk (x) ∈ W 1,1 (I) with Gk (0) = Gk (2π ), 1 ≤ k ≤ N, where N ≥ 3 and 1 ≤ m < q ≤ N − 1, m, q ∈ N. I) Let ak > 0 and ak = n, n ∈ N for 1 ≤ k ≤ m. II) Let ak = nk , nk ∈ N and e±ink x (Gk (x), √ )L2 (I) = 0 f or m + 1 ≤ k ≤ q. 2π

(13)

(Gk (x), 1)L2 (I) = 0 f or q + 1 ≤ k ≤ N.

(14)

III) Let ak = 0 and

Let Fk (u, 0) = Fk (u, 2π ) for u ∈ RN and k = 1, ..., N. We introduce the Fourier transform for periodic functions on the [0, 2π ] interval as ˆ



e−inx Gk (x) √ dx, n ∈ Z. 2π

(15)

Gk, n n2 Gk, n l∞ , l∞ }, 1 ≤ k ≤ m. |n| − ak |n| − ak

(16)

Gk, n n2 Gk, n l∞ , l∞ }, m + 1 ≤ k ≤ q. |n| − nk |n| − nk

(17)

Gk, n :=

0

and define the following expressions Pk := max{ Pk := max{

Gk, n l∞ , nGk, n l∞ }, q + 1 ≤ k ≤ N. (18) n By virtue of Lemma A3 of the Appendix of [29] under Assumption 3 the quantities given by (16), (17) and (18) are finite, which allows us to define Pk := max{

P := maxPk , 1 ≤ k ≤ N with Pk stated in formulas (16), (17) and (18). To study the existence of stationary solutions for our system we use the corresponding functional space H 2 (I) = {v(x) : I → R | v(x), v (x) ∈ L2 (I),

v(0) = v(2π ),

v (0) = v (2π )},

aiming at uk (x) ∈ H 2 (I), 1 ≤ k ≤ m. Then we introduce the following auxiliary constrained subspaces  e±ink x  = 0}, nk ∈ N, m + 1 ≤ k ≤ q, Hk2 (I) := {v ∈ H 2 (I) | v(x), √ 2π L2 (I)

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with the goal of having uk (x) ∈ Hk2 (I), m + 1 ≤ k ≤ q. And, finally H02 (I) = {v ∈ H 2 (I) | (v(x), 1)L2 (I) = 0}, q + 1 ≤ k ≤ N. Our goal is to have uk (x) ∈ H02 (I), q + 1 ≤ k ≤ N. The constrained subspaces defined above are Hilbert spaces as well (see e.g. Chapter 2.1 of [31]). The resulting space used for establishing the existence of solutions u(x) : I → RN of problem (5) will be the direct sum of the spaces mentioned above, namely q 2 2 N 2 Hc2 (I, RN ) := ⊕m k=1 H (I) ⊕k=m+1 Hk (I) ⊕k=q+1 H0 (I),

such that the corresponding Sobolev norm is given by u 2H 2 (I, RN ) := c

N

∑ { uk 2L (I) + uk 2L (I) }, 2

2

k=1

where u(x) : I → RN . Let us prove that the system of equations (5) in such case defines a map on the space mentioned above, which will be a strict contraction under given conditions. √ Theorem 2. Let Ω = I, Assumptions 1 and 3 hold and 2 π Pl < 1. Then the map τa v = u on Hc2 (I, RN ) defined by the system of equations (5) has a unique fixed point va (x) : I → RN , the only stationary solution of system (2) in Hc2 (I, RN ). This fixed point va (x) is nontrivial provided the Fourier coefficients Gk, n Fk (0, x)n = 0 for some k = 1, ..., N and some n ∈ Z. Note that the constrained subspaces Hk2 (I) and H02 (I) involved in the direct sum of spaces Hc2 (I, RN ) are such that the operators   d2 d2 2 2 − 2 − nk : Hk (I) → L (I) and − 2 : H02 (I) → L2 (I). dx dx having the Fredholm property, possess trivial kernels. Finally, we turn our attention to the studies of our problem in the layer domain, which is the product of the two spaces, such that one is the I interval with periodic boundary conditions as in the previous part of the work and another is the whole space of dimension either one or two, namely Ω = I × Rd = [0, 2π ] × Rd , d = 1, 2 and x = (x1 , x⊥ ), where x1 ∈ I and x⊥ ∈ Rd . The cumulative Laplacian in this context will be given by 2 2 Δ := ∂∂x2 + Δ⊥, where Δ⊥ := ∑ds=1 ∂ x∂2 . The corresponding Sobolev space for our problem will be H 2 (Ω, RN ) 1

⊥, s

of vector functions u(x) : Ω → RN , such that for k = 1, ..., N

uk (x), Δuk (x) ∈ L2 (Ω), uk (0, x⊥ ) = uk (2π , x⊥ ),

∂ uk ∂ uk (0, x⊥ ) = (2π , x⊥ ), ∂ x1 ∂ x1

where x⊥ ∈ Rd a.e. It is equipped with the norm u 2H 2 (Ω, RN ) =

N

∑ { uk 2L (Ω) + Δuk 2L (Ω) }. 2

2

k=1

√ Analogously to the whole space case treated in Theorem 3, the operators −Δ − ak : H 2 (Ω) → L2 (Ω) for ak ≥ 0 do not possess the Fredholm property. Let us show that system (5) in such case defines a map ta : H 2 (Ω, RN ) → H 2 (Ω, RN ), which is a strict contraction under the corresponding technical conditions stated below. Assumption 4. Let Gk (x) : Ω → R, Gk (x) ∈ W 1,1 (Ω), Gk (0, x⊥ ) = Gk (2π , x⊥ ) and Fk (u, 0, x⊥ ) = Fk (u, 2π , x⊥ ) for x⊥ ∈ Rd a.e., u ∈ RN , d = 1, 2 and k = 1, ..., N. Let N ≥ 3 and 1 ≤ m < q ≤ N − 1 with m, q ∈ N.

Vitali Vougalter, Vitaly Volpert / Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 75–86

I) Assume for 1 ≤ k ≤ m that we have nk < ak < nk + 1, nk ∈ Z+ = N ∪ {0}, x⊥ Gk (x) ∈ L1 (Ω) and √2 2 einx1 e±i ak −n x⊥ √ )L2 (Ω) = 0, |n| ≤ nk f or d = 1, (Gk (x1 , x⊥ ), √ 2π 2π einx1 eipx⊥ 2 (Gk (x1 , x⊥ ), √ )L2 (Ω) = 0, p ∈ S√ a.e., |n| ≤ nk f or d = 2. a2k −n2 2 π 2π II) Assume for m + 1 ≤ k ≤ q that we have ak = nk , nk ∈ N, x2⊥ Gk (x) ∈ L1 (Ω) and √2 2 einx1 e±i nk −n x⊥ √ )L2 (Ω) = 0, |n| ≤ nk − 1 f or d = 1, (Gk (x1 , x⊥ ), √ 2π 2π einx1 eipx⊥ 2 )L2 (Ω) = 0, p ∈ S√ a.e., |n| ≤ nk − 1 f or d = 2, (Gk (x1 , x⊥ ), √ n2k −n2 2π 2π e±ink x1 e±ink x1 )L2 (Ω) = 0, (Gk (x1 , x⊥ ), √ x⊥, s )L2 (Ω) = 0, 1 ≤ s ≤ d. (Gk (x1 , x⊥ ), √ 2π 2π III) Assume for q + 1 ≤ k ≤ N that we have ak = 0, x⊥ Gk (x) ∈ L1 (Ω) and (Gk (x), 1)L2 (Ω) = 0.

k, n (p)| ≤ k, n (p) L∞ := sup{p∈Rd , n∈Z} |G G n,p

(19) (20)

(21) (22) (23)

(24)

Let us use the Fourier transform for functions on such a product of spaces, such that ˆ ˆ 2π 1 −ipx⊥  dx⊥ e Gk (x1 , x⊥ )e−inx1 dx1 , p ∈ Rd , n ∈ Z, k = 1, ..., N. Gk, n (p) := d+1 0 (2π ) 2 Rd Hence

81

1 (2π )

d+1 2

(25)

Gk L1 (Ω) .

We define the following quantities Rk := max{

k, n (p) G

k, n (p) (p2 + n2 )G L∞n,p , L∞n,p }, k = 1, ..., m. p2 + n2 − ak p2 + n2 − ak

(26)

k, n (p) G

k, n (p) (p2 + n2 )G L∞n,p , L∞n,p }, k = m + 1, ..., q. (27) p2 + n2 − nk p2 + n2 − nk k, n (p) G k, n (p) L∞ }, k = q + 1, ..., N. L∞n,p , p2 + n2 G (28) Rk := max{ n,p p2 + n2 Assumption 4 along with Lemmas A4, A5 and A6 of the Appendix of [29] yield that the expressions given by (26), (27) and (28) are finite. This enables us to define Rk := max{

R := maxRk , k = 1, ..., N. with Rk given in (26), (27) and (28). The final proposition of our article is as follows. √ d+1 Theorem 3. Let Ω = I × Rd , d = 1, 2, Assumptions 1 and 4 hold and 2(2π ) 2 Rl < 1. Then the map ta v = u on H 2 (Ω, RN ), which is defined by the system of equations (5) admits a unique fixed point va (x) : Ω → RN , which is the only stationary solution of problem (2) in H 2 (Ω, RN ). This fixed point va (x) is nontrivial provided  that the intersection of supports of the Fourier images of functions suppF k (0, x)n (p) ∩ suppGk, n (p) is a set of d nonzero Lebesgue measure in R for some k = 1, ..., N and some n ∈ Z. Note that the maps discussed in the theorems above are applied to real valued vector functions by means of the assumptions on Fk (u, x) and Gk (x), k = 1, ..., N present in the nonlocal terms of problem (5).

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2 The System in the whole space Proof of Theorem 1. First let us suppose that when Ω = Rd , d = 1, 2, 3 there exists v(x) ∈ H 2 (Rd , RN ) such that system (5) admits two solutions u(1),(2) (x) ∈ H 2 (Rd , RN ). Thus the difference vector function w(x) := u(1) (x) − u(2) (x) ∈ H 2 (Rd , RN ) is a solution of the homogeneous system of equations √ −Δwk = ak wk , 1 ≤ k ≤ N. √ Because the −Δ operator does not have any nontrivial eigenfunctions belonging to L2 (Rd ), we obtain wk (x) = 0 a.e. in Rd for k = 1, ..., N. Let us choose arbitrarily a vector function v(x) ∈ H 2 (Rd , RN ) and apply the standard Fourier transform (9) to both sides of problem (5). This implies d

uk (p) = (2π ) 2

k (p)  fk (p) G , k = 1, ..., N. |p| − ak

(29)

Here  fk (p) stands for the Fourier image of Fk (v(x), x). We obtain the elementary estimates using expressions (10) and (11) d fk (p)| |uk (p)| ≤ (2π ) 2 Mk | 

and

d |p2 uk (p)| ≤ (2π ) 2 Mk |  fk (p)|, k = 1, ..., N.

This gives us the upper bound for the norm u 2H 2 (Rd , RN ) ≤ 2(2π )d

N

∑ Mk2 Fk (v(x), x) 2L (R ) < ∞. 2

d

k=1

by virtue of inequality (3) of Assumption 1. Therefore, for any v(x) ∈ H 2 (Rd , RN ) there exists a unique vector function u(x) ∈ H 2 (Rd , RN ), which satisfies system (5) and its Fourier image is given by (29). Hence the map Ta : H 2 (Rd , RN ) → H 2 (Rd , RN ) is well defined. This enables us to choose arbitrary v(1),(2) (x) ∈ H 2 (Rd , RN ) and obtain their images under the map u(1),(2) := (1),(2) ∈ H 2 (Rd , RN ) and derive easily the bounds for k = 1, ..., N Ta v d





(1) (2) (1) (2) |uk (p) − uk (p)| ≤ (2π ) 2 M| fk (p) − fk (p)|, d





(1) (2) (1) (2) |p2 uk (p) − p2 uk (p)| ≤ (2π ) 2 M| fk (p) − fk (p)|.

 (1),(2) (p) stand for the Fourier transforms of Fk (v(1),(2) (x), x). This yields the bound on the In this context fk corresponding norm of the difference of vector functions N

u(1) − u(2) 2H 2 (Rd , RN ) ≤ 2(2π )d M 2 ∑ Fk (v(1) (x), x) − Fk (v(2) (x), x) 2L2 (Rd ) . k=1

(1),(2)

(x) ∈ H 2 (Rd ) ⊂ L∞ (Rd ), 1 ≤ By virtue of the Sobolev embedding theorem for k = 1, ..., N we have vk d ≤ 3. Inequality (4) trivially gives us √ d Ta v(1) − Ta v(2) H 2 (Rd , RN ) ≤ 2(2π ) 2 Ml v(1) − v(2) H 2 (Rd , RN ) . The constant in the right side of this bound is less than one by means of the assumption of the theorem. Therefore, the Fixed Point Theorem implies the existence of a unique vector function va (x) ∈ H 2 (Rd , RN ), such that Ta va = va . This is the only stationary solution of system (2) in H 2 (Rd , RN ). Finally, let us assume that va (x) = 0 a.e. in Rd . This will yield the contradiction to the condition that for some k = 1, ..., N the Fourier images of Gk (x) and Fk (0, x) do not vanish simultaneously on some set of nonzero Lebesgue measure in Rd .

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3 The system on the [0, 2π ] interval Proof of Theorem 2. We first suppose that for some v(x) ∈ Hc2 (I, RN ) there exist two solutions u(1),(2) (x) ∈ Hc2 (I, RN ) of system (5) with Ω = I. Then the difference vector function w(x) := u(1) (x) − u(2) (x) ∈ Hc2 (I, RN ) will be a solution of the system of equations  d2 − 2 wk = ak wk , k = 1, ..., N. dx + Due to Assumption 3, we have ak = n, n ∈ Z = N ∪ {0} when k = 1, ..., m and as a consequence, they are not the eigenvalues of the operator −d 2 /dx2 on L2 (I) with periodic boundary conditions. Hence, wk (x) vanishes a.e. in I when k = 1, ..., m. For k = m + 1, ..., q the values of ak are identical to the nonzero eigenvalues of the square root of the negative second derivative operator with periodic boundary conditions on the [0, 2π ] 2 interval but wk belong to the constraned subspaces √ Hk (I). Thus, wk = 0 a.e. in I for k = m + 1, ..., q since ±in x k / 2π . By virtue of Assumption 3 the constants ak are zeros for they are orthogonal to the eigenfunctions e k = q + 1, ..., N. But wk belong to the constrained subspace H02 (I) of functions orthogonal to the zero mode of −d 2 /dx2 on L2 (I) with periodic boundary conditions. Thus, wk (x) vanishes a.e. in I when k = q + 1, ..., N as well. We assume that v(x) ∈ Hc2 (I, RN ) is arbitrary. Let us apply the Fourier transform (15) to both sides of the system of equations (5) considered on the interval [0, 2π ] and obtain

uk, n =





Gk, n fk, n , |n| − ak

n ∈ Z,

(30)

where fk, n := Fk (v(x), x)n . Apparently, the Fourier coefficients of the second derivatives are given by (−uk )n =





n2 Gk, n fk, n , |n| − ak

n ∈ Z.

We trivially obtain the estimate from above u 2H 2 (I, RN ) = c

N

∑{





k=1 n=−∞

|uk, n |2 +





n=−∞

|n2 uk, n |2 } ≤ 4π

N

∑ Pk2 Fk (v(x), x) 2L (I) < ∞, 2

k=1

which comes from inequality (3) of Assumption 1. Thus, for an arbitrarily chosen vector function v(x) ∈ Hc2 (I, RN ) there exists a unique u(x) ∈ Hc2 (I, RN ), which satisfies the system of equations (5) and its Fourier coefficients are given by formula (30), such that the map τa : Hc2 (I, RN ) → Hc2 (I, RN ) is well defined. Note that orthogonality relations (13) and (14) (30) yield that for k = m + 1, ..., q components uk (x) are √ along with ±in x 2 k / 2π in L (I) and for k = q + 1, ..., N functions uk (x) are orthogonal to orthogonal to Fourier harmonics e 1 in L2 (I), since the corresponding Fourier coeffients can be made equal to zero. Then we choose arbitrary vector functions v(1),(2) (x) ∈ Hc2 (I, RN ), such that their images under the map defined above are u(1),(2) := τa v(1),(2) ∈ Hc2 (I, RN ) and arrive easily at the estimate u(1) − u(2) 2H 2 (I, RN ) = c





N

∑{ ∑

|u(1) k, n − u(2) k, n |2 +





|n2 (u(1) k, n − u(2) k, n )|2 }

n=−∞ k=1 n=−∞ N Pk2 Fk (v(1) (x), x) − Fk (v(2) (x), x) 2L2 (I) . 4π k=1



(1),(2)

(x) ∈ H 2 (I) ⊂ L∞ (I) for k = 1, ..., N. Using Evidently, by means of the Sobolev embedding theorem vk (4) we easily obtain √ τa v(1) − τa v(2) Hc2 (I, RN ) ≤ 2 π Pl v(1) − v(2) Hc2 (I, RN ) .

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The constant in the right side of this bound is less than one by virtue of the assumption of the theorem. Therefore, the Fixed Point Theorem implies the existence and uniqueness of a vector function va (x) ∈ Hc2 (I, RN ), which satisfies τa va = va . This is the only stationary solution of the system of equations (2) in Hc2 (I, RN ). Finally, we suppose that va (x) vanishes a.e. in the interval I. This will imply the contradiction to our assumption that the Fourier coefficients Gk, n Fk (0, x)n = 0 for some k = 1, ..., N and some n ∈ Z. 4 The system in the layer domain Proof of Theorem 3. First of all we suppose that there exists v(x) ∈ H 2 (Ω, RN ) generating u(1),(2) (x) ∈ H 2 (Ω, RN ), which satisfy system (5). Then the difference of such vector functions w(x) := u(1) (x) − u(2) (x) ∈ H 2 (Ω, RN ) will be a solution to the homogeneous system of equations √

−Δwk = ak wk , k = 1, ..., N.

We apply the partial Fourier transform with respect to the first variable to this system and obtain 1 with wk, n (x⊥ ) := √ 2π

ˆ 0



−Δ⊥ + n2 wk, n (x⊥ ) = ak wk, n (x⊥ ), k = 1, ..., N, n ∈ Z.

wk (x1 , x⊥ )e−inx1 dx1 . Clearly, wk 2L2 (Ω) =





n=−∞

wk, n 2L2 (Rd ) .

Therefore, wk, n (x⊥ ) ∈ L2 (Rd ), k = 1, ..., N, n ∈ Z. But the operator −Δ⊥ + n2 considered on L2 (Rd ) does not have any nontrivial eigenfunctions. This implies that w(x) = 0 a.e. in Ω. Let us choose an arbitrary vector function v(x) ∈ H 2 (Ω, RN ) and apply the Fourier transform (25) to both sides of problem (5). This yields uk, n (p) = (2π )

d+1 2

k, n (p)  fk, n (p) G , k = 1, ..., N, 2 p + n2 − ak

n ∈ Z,

p ∈ Rd ,

d = 1, 2,

(31)

where fk, n (p) stands for the Fourier image of Fk (v(x), x). Apparently, for the above mentioned values of k, n and p we have the bounds in terms of the quantities given by (26), (27) and (28) as | uk, n (p)| ≤ (2π )

d+1 2

Rk |  fk, n (p)|

and

|(p2 + n2 ) uk, n (p)| ≤ (2π )

d+1 2

Rk |  fk, n (p)|.

By virtue of (3) of Assumption 1 we arrive at u 2H 2 (Ω, RN ) =

N

∑{





k=1 n=−∞

ˆ Rd

| uk, n (p)|2 d p +





n=−∞

ˆ Rd

|(p2 + n2 ) uk, n (p)|2 d p}

N

≤ 2(2π )d+1 ∑ Rk 2 Fk (v(x), x) 2L2 (Ω) < ∞. k=1

Hence, for any vector function v(x) ∈ H 2 (Ω, RN ) there exists a unique u(x) ∈ H 2 (Ω, RN ) which solves the system of equations (5) and its Fourier image is given by formula (31). Therefore, the map ta : H 2 (Ω, RN ) → H 2 (Ω, RN ) is well defined.

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We choose two arbitrary vector functions v(1),(2) ∈ H 2 (Ω, RN ) such that their images under the map discussed above are u(1),(2) := ta v(1),(2) ∈ H 2 (Ω, RN ). Hence (1)

u

− u(2) 2H 2 (Ω, RN )

N

=∑





k=1 n=−∞

ˆ Rd



2 2 2

2 (1) (2) (1) (2) d p{|u

k, n (p) − u k, n (p)| + |(p + n )(u k, n (p) − u k, n (p))| } N

≤2(2π )d+1 R2 ∑ Fk (v(1) (x), x) − Fk (v(2) (x), x) 2L2 (Ω) . k=1

(1),(2)

Obviously , by virtue of the Sobolev embedding theorem vk means of (4) we easily derive the estimate ta v(1) − ta v(2) H 2 (Ω, RN ) ≤



2(2π )

d+1 2

(x) ∈ H 2 (Ω) ⊂ L∞ (Ω) for k = 1, ..., N. By

Rl v(1) − v(2) H 2 (Ω, RN ) .

with the constant in its right side less than one due to our assumption. Therefore, the Fixed Point Theorem gives us the existence and uniqueness of a vector function va (x) ∈ H 2 (Ω, RN ), for which ta va = va holds. This is the only stationary solution of problem (2) in H 2 (Ω, RN ). Finally, we suppose that the vector function va (x) = 0 a.e. in Ω. This will contradict to the assumption of the theorem that there exists k = 1, ..., N and n ∈ Z, such that d  suppF k (0, x)n (p) ∩ suppGk, n (p) is a set of nonzero Lebesgue measure in R . 5 Discussion We will conclude the article with a brief discussion of biological interpretations of the results obtained above. All tissues and organs in a biological organism are characterized by cell distribution with respect to their genotype. Without mutations all cells would have an identical genotype. Because of mutations, the genotype changes and represents a certain distribution around its principal value. Stationary solutions of such system give stationary cell distribution with respect to the genotype. Existence of such stationary distributions is a significant property of biological organisms allowing their existence as steady state systems. Existence of stationary solutions is established in the spaces of integrable functions decaying at infinity, with periodic boundary conditions on an interval and in a mixed situation in a layer. Biologically this implies that the cell distribution with respect to the genotype decays as the distance from the principal genotype increases. The results of the article establish that conditions should be imposed on cell proliferation, mutations and influx/efflux to obtain such distributions. In the context of population dynamics, such result is applicable also to biological species which individuals are distributed around a certain average genotype. In such case, existence of stationary solutions is related to the existence of biological species (see [32]). References [1] Agranovich, M.S. (1997), Elliptic boundary problems, Encyclopaedia Math. Sci., Partial Differential Equations, IX, Springer, Berlin, 79, 1-144. [2] Lions, J.L. and Magenes, E. (1968), Problemes aux limites non homogenes et applications. Dunod, Paris, (1), 372. [3] Volevich, L.R. (1965), Solubility of boundary value problems for general elliptic systems, Mat. Sb., 68(110), 373-416; (1968), English translation: Amer. Math. Soc. Transl., 67(2), 182-225. [4] Volpert, V. (2011), Elliptic partial differential equations. Volume I. Fredholm theory of elliptic problems in unbounded domains. Birkh¨auser, 639. [5] Vougalter, V. and Volpert, V. (2012), Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems, Anal. Math. Phys., 2(4), 473-496. [6] Vougalter, V. and Volpert, V. (2011), Solvability conditions for some non Fredholm operators, Proc. Edinb. Math. Soc., (2), 54(1), 249-271.

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[7] Volpert, V., Kazmierczak, B., Massot, M., and Peradzynski, Z. (2002), Solvability conditions for elliptic problems with non-Fredholm operators, Appl. Math., 29(2), 219-238. [8] Vougalter, V. and Volpert, V. (2010), On the solvability conditions for some non Fredholm operators, Int. J. Pure Appl. Math., 60(2), 169-191. [9] Vougalter, V. and Volpert, V. (2012), On the solvability conditions for the diffusion equation with convection terms, Commun. Pure Appl. Anal., 11(1), 365-373. [10] Vougalter, V. and Volpert, V. (2010), Solvability relations for some non Fredholm operators, Int. Electron. J. Pure Appl.Math., 2(1), 75-83. [11] Volpert, V. and Vougalter, V. (2011), On the solvability conditions for a linearized Cahn-Hilliard equation, Rend. Istit. Mat. Univ. Trieste, 43, 1-9. [12] Vougalter, V. and Volpert, V. (2010), Solvability conditions for some systems with non Fredholm operators, Int. Electron. J. Pure Appl. Math., 2(3), 183-187. [13] Vougalter, V. and Volpert V. (2012), Solvability conditions for a linearized Cahn-Hilliard equation of sixth order, Math. Model. Nat. Phenom., 7(2), 146-154. [14] Ducrot, A., Marion M., and Volpert, V. (2005), Systemes de r´eaction-diffusion sans propri´et´e de Fredholm, CRAS, 340(9), 659-664. [15] Ducrot, A., Marion M., and Volpert V. (2008), Reaction-diffusion problems with non Fredholm operators, Advances Diff. Equations , 13(11-12), 1151-1192. [16] Ducrot, A., Marion, M., and Volpert, V. (2009), Reaction-diffusion waves (with the Lewis number different from 1). Publibook, Paris, 113. [17] Vougalter, V. and Volpert, V. (2011), On the existence of stationary solutions for some non-Fredholm integrodifferential equations, Doc. Math., 16, 561-580. [18] Carreras, B., Lynch, V., and Zaslavsky, G. (2001), Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8, 5096-5103. [19] Solomon, T., Weeks, E. and Swinney, H. (1993), Observation of anomalous diffusion and Levy flights in a twodimensional rotating flow, Phys. Rev. Lett., 71, 3975-3978. [20] Manandhar, P., Jang, J., Schatz, G.C., Ratner, M.A., and Hong, S. (2003), Anomalous surface diffusion in nanoscale direct deposition processes, Phys. Rev. Lett., 90, 4043-4052. [21] Sancho, J., Lacasta, A., Lindenberg, K., Sokolov, I., and Romero, A. (2004), Diffusion on a solid surface: Anomalous is normal, Phys. Rev. Lett., 92, 250601. [22] Scher, H. and Montroll, E. (1975), Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B, 12, 24552477. [23] Metzler, R. and Klafter, J. (2000), The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77. [24] Apreutesei, N., Bessonov, N., Volpert, V., and Vougalter. V. (2010), Spatial Structures and Generalized Travelling Waves for an Integro-Differential Equation, Discrete Contin. Dyn. Syst. Ser. B, 13(3), 537-557. [25] Berestycki, H., Nadin, G., Perthame, B., and Ryzhik L. (2009), The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity, 22(12), 2813-2844. [26] Genieys, S., Volpert, V., and Auger, P. (2006), Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1(1), 63-80. [27] Beck, M., Ghazaryan, A., and Sandstede, B. (2009), Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities, J. Differential Equations, 246, 4371-4390. [28] Ghazaryan, A. and Sandstede, B. (2007), Nonlinear convective instability of Turing-unstable fronts near onset: a case study, SIAM J. Appl. Dyn. Syst. 6(2), 319-347. [29] Vougalter, V. and Volpert, V. (2016), Existence of stationary solutions for some non-Fredholm integro-differential equations with superdiffusion, Preprint. [30] Shen, W. and Zhang, A. (2010), Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations, 249(4), 747-795. [31] Hislop, P.D. and Sigal, I.M. (1996), Introduction to spectral theory. With applications to Schr¨odinger operators. Springer, 337. [32] Bessonov, N., Reinberg, N., and Volpert, V. (2014), Mathematics of Darwins Diagram, Math. Model. Nat. Phenom., 9(3), 5-25.

Discontinuity, Nonlinearity, and Complexity 6(1) (2017) 87–103

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Defensive Driving Strategy for Autonomous Ground Vehicle in Mixed Traffic Xiang Li1 , Jian-Qiao Sun2† 1 2

School of Engineering, University of California at Merced, Merced, CA 95343, USA Department of Mechanics, Tianjin University, Tianjin, 300072, China Submission Info Communicated by Valentin Afraimovich Received 1 September 2016 Accepted 5 October 2016 Available online 1 April 2017 Keywords Autonomous ground vehicle Mixed traffic Defensive driving Motion planning Multi-objective optimization

Abstract One of the challenges of autonomous ground vehicles (AGVs) is to interact with human driven vehicles in the traffic. This paper develops defensive driving strategies for AGVs to avoid problematic vehicles in the mixed traffic. A multi-objective optimization algorithm for local trajectory planning is proposed. The dynamic predictive control is used to derive optimal trajectories in a rolling horizon. The intelligent driver model and lane-changing rules are employed to predict the movement of the vehicles. Multiple performance objectives are optimized simultaneously, including traffic safety, transportation efficiency, driving comfort and path consistency. The multi-objective optimization problem is solved with the cell mapping method. Different and relatively simple scenarios are created to test the effectiveness of the defensive driving strategies. Extensive experimental simulations show that the proposed defensive driving strategy is promising and may provide a new tool for designing the intelligent navigation system that helps autonomous vehicles to drive safely in the mixed traffic. ©2017 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Intelligent autonomous vehicles have great potential to improve human mobility, enhance traffic safety and achieve fuel economy [1]. However, to realize this potential, the autonomous driving technologies such as sensing, decision making and motion planning must learn how to interact with the vehicles driven by human. The mixed traffic of human-driven and autonomous vehicles has been drawing increasing attention in recent years. This paper proposes a novel multi-objective optimization algorithm for designing defensive driving strategies for autonomous ground vehicles (AGVs) to avoid problematic human-driven vehicles in the mixed traffic. The research findings are obtained from extensive computer simulations. Since the relationship between the performances and control parameters of the traffic network is highly nonlinear, complex and random, it is very common to use computer simulations in the traffic research. Driving autonomous vehicles is usually planned in four levels [2], i.e. route planning, path planning, maneuver choice and trajectory planning. Route planning is concerned with finding the best global reference path from a given origin to a destination. Path, maneuver and trajectory planning, which are often combined as one, † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2017 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2017.03.008

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provide the AGV with a safe and efficient local trajectory along the reference path considering vehicle dynamics, maneuver capabilities and road structure in the presence of other traffic [3]. A reference path is assumed to be obtained as prior information from the high-level planner in this study. We develop a feasible multi-objective optimal local trajectory for the AGV to follow the reference path in a safe manner. The low-level control is supposed to be able to perform high-accuracy tracking and to direct the vehicle on the planned trajectory. Recently, the model predictive control (MPC) which is an optimal control method applied in a rolling horizon framework, appears to be very promising and has been popularly used to solve trajectory planning problems [4, 5]. It is robust to uncertainty, disturbance and model mismatch. However, solving the optimization problem in real-time requires remarkable computational effort. To deal with the difficulties, a lot of researches have been carried out on sampling-based trajectory planning approaches. Two classes of trajectory generations are studied, i.e. control-space sampling and state-space sampling [6]. The control-space sampling method aims to generate a feasible control in the parameterized control space. The trajectories can be generated through forward simulation of the differential equations. However, the road environment must be considered in the sampling process, which may lead to discrepancy between two consecutive plans, and overshoot and oscillation in trajectory tracking process [6, 7]. On the other hand, the state-space sampling method selects a set of terminal states and computes the trajectories connecting initial and terminal states. The information of the environment can be well exploited. System-compliant trajectories can be obtained via forward simulations using the vehicle system model [8, 9]. The state-space sampling method is used in this study for trajectory generation. In the planning of autonomous vehicle trajectory, static obstacles are often considered to avoid collision. Li et al. have proposed an integrated local trajectory planning and tracking control framework for AGVs driving along a reference path with obstacle avoidance [10]. An MPC-based path generation algorithm is applied to produce a set of smooth and kinematically-feasible paths connecting the initial state with the sampled terminal states. The algorithm proposed in [11] generates a collision free path that considers vehicle dynamic constraints, road structure and different obstacles inside the horizon of view. Yoon et al. have developed a model-predictive approach for trajectory generation of AGVs combined with a tire model [12]. Information on static obstacles is incorporated online in the nonlinear model-predictive framework as they are sensed within a limited sensing range. On public roads, the influence of surrounding traffic on AGVs is significant. Trajectory planning techniques need to anticipate the behaviors of the surrounding traffic. Some trajectory planning is based on the assumption that the surrounding traffic will keep constant speed [5,13]. Shim and colleagues have proposed a motion planner that can compute a collision-free reference trajectory [14]. When the path of a moving obstacle is estimated and a possible collision in the future is detected, a new path is computed to avoid the collision. A real-time autonomous driving motion planner with trajectory optimization has been proposed by Xu and colleagues [15]. Wei et al. have proposed a motion planning algorithm considering social cooperation between the autonomous vehicle and surrounding cars [16]. Kinematically and dynamically feasible paths are generated first assuming no obstacle on the road. A behavioral planner then takes static and dynamic obstacles into account. An optimization-based path planner is presented by Hardy et al. [17] that is capable of planning multiple contingent paths to account for uncertainties in the future trajectories of dynamic obstacles. The problem of probabilistic collision avoidance is addressed for AGVs that are required to safely interact with other vehicles with unknown intentions. A human driver makes decisions on lane-changing, overtaking etc. based on his/her experience and judgement of the behavior of the surrounding vehicles. The ability to judge the surrounding vehicle behavior including the driving pattern, aggressiveness and intentions and to make adjust his/her own driving to avoid problematic vehicles is an important part of knowledge an experienced driver has. This work intents to develop a multiobjective optimization algorithm to allow AGVs to possess such an ability. Specifically, car-following and lane-changing behaviors of AGVs in straight road segments are optimized to avoid problematic vehicles. The objectives include transportation efficiency, traffic safety, driving comfort, path consistency etc. The multiobjective optimization problem (MOP) of trajectory planning is formulated and the cell mapping method [18] is

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adopted to solve the MOP. This work may provide a useful tool for intelligent navigation of AGVs in the mixed traffic. The paper starts with a brief description of the autonomous vehicle planning scheme in Section 2. The traffic model is provided in Section 3. The motion and trajectory planning algorithms are presented in Section 4 and 5, respectively. The proposed method is tested in different, but relatively simple, scenarios to verify the effectiveness in Section 6. We close the paper with conclusions in Section 7. 2 Autonomous driving architecture Figure 1 shows the architecture of the automated driving control. In the following, we discuss the architecture and the assumptions of this study.

Fig. 1 Autonomous driving architecture.

The control architecture includes a dynamic route planning module, which makes use of on-board sensors. The sensors are assumed to be able to provide real-time information of the surrounding vehicles such as positions, velocities, careless-driving, aggressiveness, etc.. The motion planning algorithm for defensive driving to avoid careless or aggressive drivers selects one of ten possible motions in Figure 2. This approach significantly reduces the online computational time in the dynamic optimization of trajectories. In the dynamic trajectory planning, the Pareto optimal set of trajectories can be obtained. Multiple performance objectives such as safety, transportation efficiency, driving comfort etc. are optimized simultaneously with the cell mapping method. All the optimal trajectories must respect the longitudinal and lateral vehicle constraints to ensure the safety, stability and controllability. The multi-objective optimization problems are solved online in a rolling horizon way to derive optimal designs for the next control horizon. The time-domain updating scheme of the control is presented in Figure 3. The prediction horizon N p and control horizon Nc are assumed to be constant. The final output is a desired trajectory representing the spatiotemporal longitudinal and lateral position, velocity and acceleration of the AGV. The prediction horizon is supposed to be much longer than the control horizon, allowing the controller to tolerate delays. It also increases the reliability of the system, since the lower-level controller always has a relatively long trajectory to execute, even if the higher-level planner stops working for some time. At the next control step, the prediction horizon is shifted one step forward, and the optimization is carried out again with the updated traffic information. The control system benefits from the rolling horizon framework with feedbacks

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Fig. 2 Options for local motion planning.

Fig. 3 Timing of dynamic predictive control scheme.

from the real traffic periodically, which makes the proposed control robust to uncertainties and disturbances. As stated earlier, the AGV is assumed to be collecting real-time information of the neighboring vehicles within the sensor range lsense , including the longitudinal and lateral positions, velocities and accelerations. In the time step tctrl,p = tctrl − tdelay where tdelay is the delayed time step reserved for online computing, the multiobjective optimization problem starts to be solved based on the current information. The optimal controller k(tctrl ) is assumed to be derived at control step tctrl where tctrl = nint × Nc and nint is an integer. The prediction horizon is the following N p steps from tctrl,p . The future traffic with a wide variety of control designs is forecasted, and the design leading to the optimal driving performance is used at the control step. Finally, we assume that the lower level cruise control of the AGV can track the desired trajectory with high accuracy. 3 Traffic models The models for traffic simulation and prediction are widely available in the literature and are briefly reviewed here. Only the straight road is considered in this study. Vehicles are assumed to travel longitudinally in a single lane and to move laterally when changing lane.

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3.1

91

Single-lane vehicle model

Car-following models that describe how drivers follow the leading vehicles have been studied for more than half a century [19, 20]. Popular models include intelligent driver model [21, 22], Gazis-Herman-Rothery model [23, 24], safety distance or collision avoidance model, Helly linear model, fuzzy logic-based model and optimal velocity model [25]. In this paper, the popular intelligent driver model (IDM) is used for traffic simulation and prediction, that has been validated and calibrated by many researchers [21, 22]. IDM describes realistically both the driving behavior of individual drivers and the collective dynamics of traffic flow such as stop-and-go waves. In IDM, the vehicle acceleration a is a continuous function of its velocity v, the spatial gap dlead to the leading vehicle and the velocity difference Δv with the leader. d ∗ (v, Δv) 2 ) ], vdes dlead vΔv , d ∗ (v, Δv) = d0 + tgap v + √ 2 amax amin a = amax [1 − (

v

)δ − (

(1) (2)

where vdes denotes the desired speed, δ represents a fixed acceleration exponent, d ∗ is the desired gap with the leader, d0 is the minimum separating distance and tgap denotes a preferred time gap. 3.2

Lane-changing model

The lane-changing model is based on the well established models in the literature [26–31]. As shown in Figure 4, for the subject vehicle, let Vehl,l and Vehl, f denote the leading and following vehicles in the left adjacent lane, respectively. Vehc,l , Vehc, f , Vehr,l and Vehr, f represent those in the current and right lane. The right adjacent lane is taken as the target lane for instance. At time t, the driver of the subject vehicle has the desire to change lane for speed advantage when the following requirements are met, v(t) < vdes , dlead (t) < D f ree , Δv(t) < Δvdi f ,min ,

(3)

where D f ree is the upper threshold of distance and Δvdi f ,min denotes a preset value of minimum speed difference.

Fig. 4 A three-lane traffic scenario.

Feasibility and safety must be checked before changing lane, such that the following inequalities must hold, dt,lead (t) > dmin , dt, f ollow (t) > dmin ,

(4)

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where dt,lead and dt, f ollow denote the longitudinal gaps between the subject vehicle and the leader and follower in the target lane, and dmin is the minimum acceptable gap. Time-to-collision (TTC) is a safety indicator that represents the time for two vehicles to collide if they remain at the current longitudinal speed without lane-changing. A minimum time-to-collision tsa f e is required for safety to change lane such that, (5) tt,lead (t) > tsa f e , tt, f ollow (t) > tsa f e , where tt,lead and tt, f ollow denote the longitudinal TTCs between the subject vehicle and the leader and follower in the target lane. In addition, the target lane has to provide enough space for the subject vehicle to accelerate to higher speed. The following condition must hold, dt,lead (t) > v(t)textra ,

(6)

where textra is the required minimum time headway. We also impose a minimum time tmin between two consecutive lane-changings for one vehicle in order to prevent frequent lane-changings. The duration of lane-changing sometimes is ignored and the lane-changing process is regarded as an instantaneous movement in some models [26,30,32]. However, the influence of lane-changing duration on the traffic is significant and cannot be disregarded [31]. In this study, each vehicle Vehi is assumed to have a pre-determined lane-changing time ti,lc that is a random number uniformly distributed in an interval [tlc,min ,tlc,max ] where tlc,max and tlc,min denote the maximum and minimum lane-changing time. In the lane-changing process, the lateral velocity is assumed to be constant while the longitudinal movement follows the IDM. For the two leading vehicles in the current and target lanes, the one with shorter longitudinal distance from the subject vehicle is considered the preceding vehicle in the car-following model. 4 Defensive motion planning The defensive motion planning provides strategies for the AGV to deal with the surrounding traffic. As Figure 2 shows, ten motion patterns are proposed to reduce the online computational time in the trajectory optimization. In the longitudinal direction, three regimes are defined, i.e. following, slowing down and speeding up. Equation (7) shows nine motions as combinations of three longitudinal moves and three lane-manuveours (changing to the left lane, staying in the current lane and changing to the right lane). The tenth move is an emergency braking motion to avoid collision. ⎧ Change to the left lane and slow down, i = 1, ⎪ ⎪ ⎪ ⎪ Stay in the current lane and slow down, i = 2, ⎪ ⎪ ⎪ ⎪ Change to the right lane and slow down, i = 3, ⎪ ⎪ ⎪ ⎪ Change to the left lane and follow, i = 4, ⎪ ⎪ ⎨ Stay in the current lane and follow, i = 5, Motion(i) = (7) Change to the right lane and follow, i = 6, ⎪ ⎪ ⎪ ⎪ ⎪ Change to the left lane and speed up, i = 7, ⎪ ⎪ ⎪ ⎪ Stay in the current lane and speed up, i = 8, ⎪ ⎪ ⎪ ⎪ Change to the right lane and speed up, i = 9, ⎪ ⎩ Emergency brake, i = 10. The defensive motion planning algorithm is presented in Table 1 to avoid careless and aggressive drivers. Note that the emergency braking occurs if the gap between the AGV and its leader is shorter than a safe distance dsa f e . Careless-driving has significant influence on traffic [33], which includes vehicle waving, close cut-in, frequent lane-changings etc. In this study, only the vehicle waving behavior is considered. In the defensive driving, the AGV is supposed to avoid the waving vehicle Vehwav in the neighborhood if the waving amplitude

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93

Table 1 Motion planning module of autonomous driving process. Motion Planning Algorithm Input: Surrounding vehicle information, Output: Desired motion plans Md , 1: if dlead < dsa f e 2:

Md ← Motion(10)

3: elseif Vehwav .Amp ≥ dwave 4:

if Vehwav = Vehl,l Md ← Motion(2) AND Motion(6)

5: 6:

else if Vehwav = Vehl, f Md ← Motion(8) AND Motion(6)

7: 8:

else if Vehwav = Vehc,l Md ← Motion(2) AND Motion(4) AND Motion(6)

9: 10:

else if Vehwav = Vehc, f Md ← Motion(8) AND Motion(4) AND Motion(6)

11: 12:

else if Vehwav = Vehr,l Md ← Motion(2) AND Motion(4)

13: 14:

else if Vehwav = Vehr, f Md ← Motion(4) AND Motion(6)

15: 16

end

17: elseif Vehagg .Agg ≥ cagg if Vehagg = Vehc,l

18: 19:

Md ← Motion(2)

20:

else if Vehagg = Vehc, f Md ← Motion(4) AND Motion(6)

21: 22:

end

23: elseif speed demand is satisfied 24:

Md ← Motion(7) AND Motion(9)

25: else 26:

Md ← Motion(5)

27: end

Vehwav .Amp exceeds a critical value dwave . We assume that Vehwav can be detected by the on-board sensors of the AGV. The defensive planner also suggests not to follow the leader closely if the leader is aggressive. If the follower shows remarkable aggressiveness, the defensive planner suggests to change lane to provide courtesy and avoid the aggressive driver. The vehicle aggressiveness Veh.Agg is quantified by the detected maximum absolute acceleration. Let Vehagg represent the more aggressive driver between the leader and follower, when Vehagg .Agg exceeds a critical value cagg , a defensive-driving motion will be followed through. If the speed advantage is demanded (see Equation (3)) and lane-changing feasibilities are satisfied (see Equations (4), (5) and (6)), the AGV will attempt to change lane to overtake. The AGV either speeds up if no leading vehicle is ahead, or follows the leader. The minimum time tmin between two consecutive lane-changings is also imposed to prevent frequent lane-changings. When lane-changing is attempted but not feasible, the longitudinal motion will follow through.

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5 Trajectory planning Once a defensive driving move is decided, short-term trajectories will be generated dynamically. A novel multiobjective optimization algorithm is proposed to find the optimal trajectory. 5.1

Trajectory formulation

The local trajectory is generated for each prediction horizon. Let t1 and t2 denote the initial and terminal time in one prediction horizon, and x0 , v0 and a0 denote the initial longitudinal position, velocity and acceleration when the AGV is at (x, y) as shown in Figure 4. v1 and a1 represent the terminal velocity and acceleration. Hence, ˙ 1 ) = v0 , x(t ¨ 1 ) = a0 , x(t1 ) = x0 , x(t ¨ 2 ) = a1 . x(t ˙ 2 ) = v1 , x(t

(8)

Note that x(t2 ) is unspecified and the velocity and acceleration v1 and a1 can fully specify the local trajectory. For this reason, we take v1 and a1 as the design parameters of the local trajectory. In the lateral direction, when lane-changing is considered, the AGV is assumed to complete the lanechanging in a pre-determined time tlc . Let y0 and y1 denote the lateral positions of the center axises of the current and target lanes respectively. ˙ 1 ) = 0, y(t ¨ 1 ) = 0, y(t1 ) = y0 , y(t ˙ 1 + tlc ) = 0, y(t ¨ 1 + tlc ) = 0. y(t1 + tlc ) = y1 , y(t

(9)

The above initial and terminal conditions indicate that the trajectory x(t) can be fourth-order polynomials of time in the interval [t1 ,t2 ] [34]. A fifth-order polynomial of time for y(t) is needed to describe the lateral movement in the time interval [t1 ,t3 ] where t3 = t1 + tlc . 4

5

i=0

i=0

x(t) = ∑ ait i , y(t) = ∑ bit i .

(10)

[x0 , v0 , a0 , v1 , a1 ]T = Mx AT ,

(11)

Hence, we have

T

T

[y0 , 0, 0, y1 , 0, 0] = My B , where AT = [a4 , a3 , a2 , a1 , a0 ],

(12)

T

B = [b5 , b4 , b3 , b2 , b1 , b0 ], ⎡ Mx =

t14 ⎢ 4t 3 ⎢ 12 ⎢ 12t ⎢ 1 ⎣ 4t 3 2 12t22

t13 3t12 6t11 3t22 6t21





t15 1 ⎢ 5t14 ⎢ 3 0⎥ ⎥ ⎢ 20t 1 ⎥ 2 0 0 ⎥ , My = ⎢ ⎢ t5 1 3 ⎢ 2t2 1 0 ⎦ ⎣ 5t 4 3 2 00 20t33 t12 t11 2t11 1

t14 4t13 12t12 t34 4t33 12t32

t13 3t12 6t11 t33 3t32 6t31

t12 t11 2t11 1 2 0 t32 t31 2t31 1 2 0

⎤ 1 0⎥ ⎥ 0⎥ ⎥. 1⎥ ⎥ 0⎦ 0

The trajectory in one prediction horizon can be created by solving Equations (11) and (12).

(13)

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5.2

95

Multi-objective optimization

As noted before, we consider the design vector k = [v1 , a1 ]T for optimizing the local trajectories. The following constraints on the design parameters are imposed. 0 ≤ x(t) ˙ ≤ vmax ,

(14)

amin ≤ x(t) ¨ ≤ amax , where ∀t ∈ [t1 ,t2 ], amax > 0 and amin < 0 denote the longitudinal maximum acceleration and deceleration, and vmax represents the maximum speed of the AGV subject to the legal speed limit. When lane-changing is attempted, the lateral trajectory can be uniquely determined based on the lane structure and corresponding constraints. The control objectives of the AGV in the defensive driving include 1) Safety, 2) Transportation efficiency, 3) Driving comfort and fuel economy, and 4) Path consistency. The solutions of a multi-objective optimization problem (MOP) form a set in the design space called the Pareto set, and the corresponding objective evaluations are the Pareto front [35]. The safety indicator is defined as the sum of the time headway tth (t) and time-to-collision (TTC) tttc (t). Time headway is the elapsed time between the front of the leading vehicle passing a point on the roadway and the front of the following vehicle passing the same point. TTC represents the time before two consecutive vehicles collide if they remain their current longitudinal speed. Let Gs f denote the safety, ˆ Gs f = tth (t) = tttc (t) =

t2

t1

(αth tth (t) + αttctttc (t))dt,

v(t) , dlead (t)

v(t)−vleader (t) , dlead (t)

0,

(15) (16)

v(t) > vleader (t), v(t) ≤ vleader (t).

(17)

where vleader is the speed of the leader, and αth and αttc represent the weights of the two items. When safety is ensured, the AGV is suggested to travel longer distance in one horizon. This distance is taken as the measure of the transportation efficiency denoted as Gte0 , Gte0 = x(t2 ).

(18)

The objective Gdc of driving comfort and fuel consumption is measured by the longitudinal acceleration and jerk of the vehicle [36–38], given by ˆ t2 (ωa a2 (t) + ω j a˙2 (t))dt. (19) Gdc = t1

˙ where ωa and ω j denote the weighting factors of a(t) and a(t). The inconsistency between the consecutive plans can result in sharp steerings, control overshoots or even instability [10]. The path consistency in the trajectory re-planning process is imposed in this study. The objective Gcon penalizes the inconsistency between the current and previous longitudinal trajectories, defined as, ˆ t2 (x(t) − x pre (t))2 dt. (20) Gcon = t1

The multi-objective optimal local trajectory problem is formulated as, min{Gs f , Gcon , Gdc , Gte }, k∈Q

(21)

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where Q is a bounded domain of the design parameters, Gte = Gte,max − Gte0 and Gte,max > 0 is a large number such that Gte is positive. Furthermore, we impose the constraints on tth (t) and tttc (t) to keep the driving performance within an acceptable range. maxtth (t) ≤ tth,lim ,

(22)

maxtttc (t) ≤ tttc,lim , where tth,lim and tttc,lim are the pre-selected upper limits. 5.3

Cell mapping method

The simple cell mapping (SCM) method, proposed by Hsu [18] and further developed by Sun and his group [39], is used to solve the MOP. The cell mapping method is well suited to search for the solutions of the MOP [39–41]. The method proposes to discretize the continuum design space into a collection of boxes or cells and to describe the searching process by cell-to-cell mappings in a finite region of interest in the design space. The SCM accepts one image cell for a given pre-image cell. The SCM can be symbolically expressed as zk+1 = C[zk ] where k is the iteration step, zk is an integer representing the cell where the system resides at the kth step, and C[·] is the integer mapping constructed from an optimization search strategy. The region out of the domain of the computational interest is called the sink cell. If the image of a cell is out of the domain of interest, we say that it is mapped to the sink cell. The sink cell always maps to itself. The SCM can be implemented initially with relatively large cells. The cyclic groups of the cells in the SCMs form a set covering the Pareto solution of the MOP. The cells in the cyclic groups can be sub-divided to improve the accuracy of the MOP solutions [39]. 5.4

Design optimization

The bounded design space is defined as 

Q = k ∈ R2 | [vlb , alb ] ≤ k ≤ [vub , aub ] .

(23)

where the lower and upper bounds [vlb , alb ] and [vub , aub ] are specified for four different driving patterns of the AGV under consideration. 1. Following alb = −a f , aub = a f ,

(24)

vlb = max(vlead,p (t2 ) − v f , 0), vub = min(vlead,p (t2 ) + v f , vdes ). 2. Slowing down alb = −ac , aub = 0,

(25)

vlb = max(v(t1 ) − ac N p , 0), vub = v(t1 ), 3. Speeding up alb = 0, aub = ac , vlb = v(t1 ), vub = min(v(t1 ) + ac N p , vdes ),

(26)

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97

4. Emergency braking

alb = amin , aub = 0,

(27)

vlb = max(v(t1 ) + amin N p , 0), vub = v(t1 ). where a f and ac denote the pre-determined acceleration fluctuating amplitudes, v f represents the velocity fluctuating amplitude, and vlead,p (t2 ) is the predicted speed of the leading vehicle at time t2 . Appropriate values of a f , ac and v f are determined based on the experimental results. The local motion planning algorithm is listed in Table 1. At each step, we decide if the AGV to change to the left lane, or to stay in the current lane, or to change to the right lane. For each decision, an MOP is solved. The Pareto sets of the three MOPs are combined to form a single Pareto set. The optimal trajectory is chosen from this Pareto set by following the algorithm in Section 5.5. In the following numerical studies, we have used a 5 × 5 × 5 coarse partition of the design space. One refinement of 3 × 3 × 3 is applied. 5.5

Selection of optimal designs

To facilitate the user to pick up an optimal design from the Pareto set to implement, we propose an algorithm that operates on the Pareto front. Let fi,min denote the minimum of the ith objective in the Pareto front and fi,max be the corresponding maximum. Define a vector, Fideal = [ f1,min , f2,min , ..., fnob j ,min ].

(28)

where Fideal is considered to be an ideal point in the objective space. Let ns denote the number of Pareto solutions. To eliminate the effect of different scale of the objectives, the Pareto front is normalized as, fi, j − fi,min , f¯i, j = fi,max − fi,min

(29)

1 ≤ i ≤ ns , 1 ≤ j ≤ nob j , where fi, j is the ith objective of the jth optimal solution, and f¯i, j represents the normalized value of fi, j . The normalized objective vector F¯i reads, (30) F¯i = [ f¯1,i , f¯2,i , ..., f¯nob j ,i ], and its norm is denoted as ri = F¯i . Let rmax and rmin denote the maximum and minimum of the norm of the points in the Pareto front. Let n per denote the percentage of control designs in the Pareto set such that their corresponding norms in the Pareto front are among the n per percent smallest ones. We refer these designs as the top n per percent. A special sub-set of top designs consists of the so-called knee points [40]. They are defined as, kknee = {ki | i = min ri }. 1≤i≤ns

(31)

It should be noted that there are different ways to normalize the Pareto front. Furthermore, the normalization affects the classification of the designs. The normalization used in this paper is intuitive and simple to implement. 6 Experimental results To verify the effectiveness of the proposed multi-objective optimization algorithm for local trajectory planning, we have tested the algorithm in different scenarios. The simulations in this study are implemented in Matlab. The sampling time step is 0.1s. The simulation parameters are listed in Table 2.

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Table 2 Parameters used in this paper. Parameter tlc

Value 4s

amax

5m2 /s

amin

−8m2 /s

Parameter

Value

Parameter

lsense

150m

dmin

2m

vf

2m/s

tsa f e

2.5s

0.8s

textra

1.5s

tgap

Np

6s

D f ree

100m

tmin

Nc

1s

Δvdi f ,min

2m/s

Gte,max

0.5s

ωa

tlc,max

5

tlc,min

3

tdelay

1

tth,lim

ωj

1

tttc,lim

αth

0.1

af

1m2 /s

αttc

ac

2m2 /s

D

cagg

Value

5s 3000m 1s 0.7s 3m2 /s

1

dsa f e

15m

8veh/km

dwave

0.4m

The best way to demonstrate the defensive driving strategies is to use the animation of the AGV with the mixed traffic in the presence of human-driven vehicles. Due to the limitation of the printed publication, we present a few special cases to show how the AGV responds to different situations. We first show how the AGV responses to a cut-in scenario. As Figure 5 shows, the vehicle Veh1 attempts to cut in the gap between the AGV VehA and the leader Veh2 when the AGV is accelerating to follow the leader. The longitudinal position and velocity of the AGV are presented in Figure 6. It is observed that the AGV slows down first to cooperate with Veh1 once its lane-changing intention is detected. The gap between VehA and Veh1 extends to the safe regime to allow to finish the lane-changing. Afterwards, the AGV speeds up again to follow the new leader. The applied speed profile is optimal in consideration of safety, driving comfort, traffic efficiency and path consistency simultaneously. More numerical results and comparisons with other non-optimal designs are provided in the following examples.

Fig. 5 The vehicle trajectories in the cut-in scenario.

Next, we demonstrate the ability of the AGV to avoid waving vehicles. Figure 7 shows that vehicle Veh3 is waving. At the time indicated by number 1, the AGV detects that the waving amplitude of Veh3 is larger than the safety level, and attempts to change to the right lane to avoid it. The movement profile is optimized considering

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99

11 10

x (m)

v (m/s)

2600

2500

9 8 7

2400

50

60

6

70

50

60

70

Time (s)

Fig. 6 The longitudinal position and velocity of the autonomous vehicle in the cut-in scenario.

Fig. 7 The vehicle trajectories when the AGV is avoiding a waving vehicle. The numbers in the vehicle blocks denote the time sequence. Other surrounding vehicles are not shown for clarity. 200

1585

150

Gte

Gdc

1580 1575 1570

100 50

2

4

6

0

8

2

4

500

500

400

400

300 200 100 1570

6

8

Gsf

Gcon

Gcon

Gsf

300 200

1575

1580

Gte

1585

100

0

50

100

150

200

Gdc

Fig. 8 The Pareto front in a longitudinal following scenario (Motion(5)). The red point denotes the selected design.

the interactions with all other vehicles nearby, which are not shown in the figure. The optimal trajectory is obtained with the best compromise over the four performance metrics. In the following, a realistic road section is taken to test the performance of the proposed algorithm. The numerical experiment is carried out in a one-way three-lane road section of 3000 meters with a periodic boundary as shown in Figure 4. We denote the vehicle density, i.e. vehicles per kilometer per lane, as D. A random number

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of vehicles with different length and desired speed are considered. 20% of all the vehicles are assumed to be waving, and the maximum waving amplitudes of them are randomly determined in the range (0, 0.8] meters. The simulations are carried out over 1000s including a 500s to let the transient effect die out. Other parameters are presented in Table 2. Due to the complex interactions with the surrounding vehicles, different motion decisions are made in different situations throughout the simulation. Figure 8 shows the Pareto front obtained in the longitudinal carfollowing scenario (Motion(5)) at time 804s. The conflicting nature between different objectives is observed. The design with the balanced compromises among the four objectives is selected by applying the algorithm in Section 5.5. As the simulation continues to time 932s, the following vehicle of the AGV in the current lane is detected to be very aggressive. Tthe planner attempts to change lane to provide courtesy (Motion(4) and Motion(6)). It is not feasible to change to the left lane at that time, and the right lane is available. The Pareto front is presented in Figure 9 and the optimal trajectory is obtained. 200 100

Gdc

Gte

195 190

50

185 180

0 0

2

4

6

8

0

2

400

400

300

300

200 100 0 180

4

6

8

Gsf

Gcon

Gcon

Gsf

200 100

185

190

Gte

195

200

0

0

50

100

Gdc

Fig. 9 The Pareto front when the AGV is attempting changing to the right lane (Motion(6)). The red point denotes the selected design.

When the simulation reaches time 969s, the following vehicle of the AGV in the left lane is sensed to be waving. The planner considers speeding up in the current lane (Motion(8)) and changing to the right lane (Motion(6)) at the same time to avoid. Figure 10 shows the combined Pareto fronts with respect to the two motions. It is seen that the blue Pareto front lies on the left side of the orange Pareto front in the two upper figures. This suggests that a higher safety can be achieved by staying in the current lane than lane-changing. No significant difference in the other objectives is observed with the two motions. Consequently, the optimal trajectory design of staying in the current lane is implemented. 7 Conclusions We have presented a multi-objective optimal local trajectory planning algorithm for defensive driving of autonomous ground vehicles. We have proposed to use the longitudinal terminal velocity and acceleration as the design parameters for the trajectory planning. Different driving performance objectives are optimized simultaneously including traffic safety, transportation efficiency, driving comfort, path consistency. The optimal Pareto solutions are obtained with the cell mapping method. An algorithm is proposed to assist the user to select and

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101

30

165

Gdc

Gte

20 160

155

10 0

1

2

3

4

1

2

250

250

200

200

150 100

4

150 100

50 155

3

Gsf

Gcon

Gcon

Gsf

50 160

Gte

165

0

5

10

15

20

25

Gdc

Fig. 10 The Pareto fronts when the AGV is considering speeding up in the current lane (Motion(8) and denoted by the blue points) and changing to the right lane (Motion(6) and denoted by the orange points). The red point denotes the selected design.

implement the optimal designs. Extensive numerical simulations show that the proposed motion and trajectory planning algorithms are promising and may provide a new tool for designing the intelligent navigation system that helps AGVs to drive safely in the mixed traffic with erratic human drivers. Acknowledgements The material in this paper is based on work supported by grants (11172197, 11332008 and 11572215) from the National Natural Science Foundation of China, and a grant from the University of California Institute for Mexico and the United States (UC MEXUS) and the Consejo Nacional de Ciencia y Tecnolog´ıa de M´exico (CONACYT) through the project “Hybridizing Set Oriented Methods and Evolutionary Strategies to Obtain Fast and Reliable Multi-objective Optimization Algorithms”. The first author would like to thank the China Scholarship Council (CSC) for sponsoring his study in the United States of America. References [1] Burns, L.D. (2013), Sustainable mobility: A vision of our transport future, Nature, 497(7448), 181–182. [2] Varaiya, P. (1993), Smart cars on smart roads: problems of control, IEEE Transactions on Automatic Control, 38(2), 195–207. [3] Katrakazas, C., Quddus, M., Chen, W.H., and Deka, L. (2015), Real-time motion planning methods for autonomous onroad driving: State-of-the-art and future research directions, Transportation Research Part C: Emerging Technologies, 60, 416–442. [4] Ziegler, J., Bender, P., Schreiber, M., Lategahn, H., Strauss, T., Stiller, C. Dang, T., Franke, U., Appenrodt, N., Keller, C.G., Kaus, E., Herrtwich, R.G., Rabe, C., Pfeiffer, D., Lindner, F., Stein, F., Erbs, F., Enzweiler, M., Knoppel, C., Hipp, J., Haueis, M., Trepte, M., Brenk, C., Tamke, A., Ghanaat, M., Braun, M., Joos, A., Fritz, H., Mock, H., Hein, M., andZeeb, E. (2014), Making Bertha drive 2014 – An autonomous journey on a historic route, IEEE Intelligent Transportation Systems Magazine, 6(2), 8–20. [5] Ziegler, J., Bender, P., Dang, T., and Stiller, C. (2014), Trajectory planning for Bertha - A local, continuous method, in: Proceedings of IEEE Intelligent Vehicles Symposium, 450–457. [6] Howard, T.M., Green, C.J., Kelly, A., andFerguson, D. (2008), State space sampling of feasible motions for highperformance mobile robot navigation in complex environments, Journal of Field Robotics, 25(6–7), 325–345.

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Aims and Scope The interdisciplinary journal publishes original and new results on recent developments, discoveries and progresses on Discontinuity, Nonlinearity and Complexity in physical and social sciences. The aim of the journal is to stimulate more research interest for exploration of discontinuity, complexity, nonlinearity and chaos in complex systems. The manuscripts in dynamical systems with nonlinearity and chaos are solicited, which includes mathematical theories and methods, physical principles and laws, and computational techniques. The journal provides a place to researchers for the rapid exchange of ideas and techniques in discontinuity, complexity, nonlinearity and chaos in physical and social sciences. Topics of interest include but not limited to • • • • • • • • • • • • • •

Complex and hybrid dynamical systems Discontinuous dynamical systems (i.e., impulsive, time-delay, flow barriers) Nonlinear discrete systems and symbolic dynamics Fractional dynamical systems and control Stochastic dynamical systems and randomness Complexity, self-similarity and synchronization Complex nonlinear phenomena in physical systems Stability, bifurcation and chaos in complex systems Turbulence and other complex phenomena in hydrodynamics Nonlinear waves and solitons Dynamical networks Combinatorial aspects of dynamical systems Biological dynamics and biophysics Pattern formation, social science and complexization

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An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 6, Issue 1

March 2017

Contents Potential Symmetries, Lie Transformation Groups and Exact Solutions of KdvBurgers Equation XiaoMin Wang, Sudao Bilige, YueXing Bai.............................................................

1-9

Conservation Laws in Group Analysis of Gas Filtration Model S.V. Khabirov……..………….….…………………………..……………………..

11-17

Existence of Semi Linear Impulsive Neutral Evolution Inclusions with Infinite Delay in Frechet Spaces Dimplekumar N. Chalishajar, K. Karthikeyan, A. Anguraj………………...……...

19-34

Wave Collision for the gKdV-4 equation. Asymptotic Approach Georgy Omel’yanov…………….…………..……......…..............…....…………..

35-47

P-Moment Exponential Stability of Caputo Fractional Differential Equations with Random Impulses Ravi Agarwal, Snezhana Hristova, Donal O’Regan…………………...…...…..….

49-63

An Analytic Technique for the Solutions of Nonlinear Oscillators with Damping Using the Abel Equation A.Ghose-Choudhury, Partha Guha..……………………….……..……….…….....

65-74

On the Existence of Stationary Solutions for Some Systems of Non-Fredholm Integro-Differential Equations with Superdiffusion Vitali Vougalter, Vitaly Volpert……………………...………………………….....

75-86

Defensive Driving Strategy for Autonomous Ground Vehicle in Mixed Traffic Xiang Li, Jian-Qiao Sun………………….……………...…..…………….…….....

87-103

Available online at https://lhscientificpublishing.com/Journals/DNC-Download.aspx

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