Traveling wave solutions to a reaction-diffusion equation | SpringerLink

Z. angew. Math. Phys. 60 (2009) 756–773 0044-2275/09/040756-18 DOI 10.1007/s00033-008-8092-0 c 2009 Birkh¨ ° auser Verlag, Basel

Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP

Traveling wave solutions to a reaction-diffusion equation Zhaosheng Feng∗ , Shenzhou Zheng and David Y. Gao

Abstract. In this paper, we restrict our attention to traveling wave solutions of a reactiondiffusion equation. Firstly we apply the Divisor Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then through this first integral, we reduce the reaction-diffusion equation to a first-order integrable ordinary differential equation, and a class of traveling wave solutions is obtained accordingly. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. We clarify the errors and instead give a refined result in a simple and straightforward manner. Mathematics Subject Classification (2000). 02.30.Jr, 84.40.Fe, 04.20.Jb. Keywords. Traveling waves, first integral, Fisher equation, Divisor theorem, autonomous system, elliptic function.

1. Introduction The problems of the propagation of nonlinear waves have fascinated scientists for over two hundred years. The modern theory of nonlinear waves, like many areas of mathematics, had its beginnings in attempts to solve specific problems, the hardest among them being the propagation of waves in water. There was significant activity on this problem in the 19th century and the beginning of the 20th century, including the classic work of Stokes, Lord Rayleigh, Korteweg and de Vries, Boussinesque, Bénard and Kolmogorov et al. to name some of the better remembered examples [29, 42]. One particularly noteworthy contribution was the explosion of activity unleashed by the numerical discovery of the soliton by Zabusky and Kruskal in the early sixties, and the earliest theoretical explanation by Gardner, Greene, Kruskal, and Miura in the later part of that decade [30], which subsequently led to the present-day theory of integrable partial differential equations. There are many phenomena in mechanical engineering and biology where a key ∗ Corresponding

author

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Traveling wave solutions to a reaction-diffusion equation

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element or precursor of a developmental process seems to be the appearance of a traveling wave of chemical concentration (or mechanical deformation). When reaction kinetics and diffusion are coupled, traveling waves of chemical concentration can effect a biochemical change much faster than straight diffusional processes. This usually gives rise to reaction-diffusion equations which in one dimensional space can look like ∂u ∂2u = k0 2 + f (u), (1) ∂t ∂x for a chemical concentration u, where k0 is the diffusion coefficient, and f (u) represents the kinetics. The classic and simplest case of nonlinear reaction-diffusion equations is that when f (u) is a quadratic nonlinearity, i.e., ∂2u ∂u = + µ + αu − βu2 , µ, α, β ∈ R. (2) ∂t ∂x2 When µ = 0, it is the so-called Fisher equation, suggested by Fisher [21] as a deterministic version of a stochastic model for the spatial spread of a favored gene in a population, but the discovery, investigation and analysis of traveling waves in chemical reactions was first presented by Luther [33, 38]. In the last century, equation (2) has became the basis for a variety of models for spatial spread, for example, in logistic population growth models [11], flame propagation [22, 41], neurophysiology [39], autocatalytic chemical reactions [6, 19], branching Brownian motion processes [10], gene-culture waves of advance [7], the spread of early farming in Europe [5], and nuclear reactor theory [12]. The seminal and classical references for traveling waves of equation (2) are that by Kolmogorov, Petrovsky and Piscunov [31], Mckean [34], Ablowitz and Zeppetella [2], Aronson and Weinberger [6], Fife [20] and Britten [11]. One approach for finding traveling wave solutions of bistable medium, introduced by Rinzel and Keller [37], is to replace the nonlinear term in (2) by a piecewise-linear approximation that retains the essential features of the reaction term: two roots connected by a continuous function with its unique maxima located between the roots. The first explicit form of the traveling wave solution for the Fisher equation was obtained by Ablowitz and Zeppetella [2] using the Painlev´ √ e analysis. The kink wave propagates from left to right with a speed v = 5/ 6. The problem of selection of appropriate speed were discussed in [23, 35, 36]. Fronts initiated on a compact support evolve to minimum velocity solutions [35]. Traveling wave solutions of equation (2) with the velocity v > 2 are stable [34], and traveling waves with v < 2 also exist but they are considered to be physically unrealistic since u becomes negative at the leading edge with decreasing oscillations about u = 0. A full discussion of real traveling waves of this equation and an extensive bibliography can be seen in [11, 20, 36, 40]. Complex traveling wave solutions including some heterclinic and homoclinic solutions of equation (2) were obtained by Guo and Chen [25] using the expanded Painlevé analysis. A discrete algorithm was introduced to solve equation (2) and predict long-time traveling wave behaviors by Zhao and Wei [43]. Approximate expressions of proper traveling waves were

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Z. Feng, S. Zheng and D. Gao

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described by Feng and Chen [16] using the Hardy Theorem. Exact solutions of equation (2) and high-order cases were established by Clarkson and Mansfield [13] using the classical and nonclassical symmetry method. Analysis of the convergence of asymptotically uniformly traveling pulled fronts and a universal relaxation behavior as time t → ∞ to equation (1) were shown by Ebert and Saarloos [15]. The technique was also generalized to more general (sets of) partial differential equations with higher spatial or temporal derivatives. For more information on recent progress in this direction we refer readers to [13, 15, 40] and references therein. Seeking innovative methods to find traveling wave solutions of nonlinear equations has been a difficult and challenging task in the field of differential equations and dynamical systems. Special types of traveling waves are of fundamental importance to our understanding of physical, chemical and biological phenomena [27, 36]. The last century has seen many ingenious techniques for obtaining traveling wave solutions of differential equations mainly through geometrical analysis and analytical studies: the inverse scattering transform [3], the Hirota method [26], Painlevé analysis [28], Cole–Hopf transform [29], Lie Symmetry [8], Lamb’s ansatz [32], etc. However, most work only for a wide class of problems. In many cases, it is not always possible and sometimes not even advantageous to express traveling waves of nonlinear differential equations explicitly and directly in terms of elementary functions through these methods, but sometimes it is possible to find elementary functions that are constant on solution curves, that is, elementary first integrals. These first integrals allow us to occasionally deduce properties that an explicit solution would not necessarily reveal. In the present paper, we focus our attention on traveling wave solutions of equation (2), whose velocity may be complex numbers. We start our study with seeking the first integral for equation (2) under certain parametric condition. The paper is organized as follows. In Section 2, we present a quasi-polynomial first integral of equation (2) by using the Divisor Theorem for two variables in the complex domain based on the ring theory of commutative algebra. In Section 3, through the obtained quasi-polynomial first integral, equation (2) is reduced to a first-order integrable ordinary differential equation. A class of traveling wave solution is accordingly derived by solving this first-order differential equation. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. In Section 4 we discuss properties of our solutions. Section 5 is a brief conclusion

2. First integral Assume that equation (2) has the traveling wave solution of the form u(x, t) = u(ζ),

ζ = x − vt + ζ0 ,

(3)

where v is wave velocity, and ζ0 is an arbitrary constant. Substituting (3) into equation (2) gives

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u00 (ζ) − ru0 (ζ) − au2 − bu − d = 0,

759

(4)

where r = −v, a = β, b = −α and d = −µ. Let z = u, y = uζ , then equation (4) is equivalent to an autonomous system ½ z˙ = P (z, y) = y, (5) y˙ = Q(z, y) = ry + az 2 + bz + d, y) where an overdot denotes differentiation with respect to ζ. Since ∂P (z, + ∂z ∂Q(z, y) = r, when r 6= 0, by virtue of the Poincaré–Bendixson Theorem [16], ∂y system (5) has no closed orbit in the Poincaré phase plane. This implies that equation (4) has neither the bell-profile solitary wave solution, nor the periodic traveling wave solution. Moreover, when b2√− 4ad > 0, equation (5)√has two 2 −4ad 2 −4ad b b homogeneous stationary states: Q(0, − 2a − b 2a ) and P (0, − 2a + b 2a ). A kink-profile traveling wave solution of system (5) describes a constant-velocity front of transition from one homogeneous state to another, which corresponds to the trajectory L(P, Q) in Figure 1 [18]. All traveling wave solutions corresponding to other trajectories (except (L(P, Q)) are unbounded.

Figure 1. The global structure of system (5) under the parametric conditions a > 0 and

b2 − 4ad > 0.

Consider the two-dimensional autonomous system (5) and its associated differential operator ∂ ∂ + Q(z, y) . X = P (z, y) ∂z ∂y

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We say that the polynomial system (5) is integrable in Liouville’s sense [24], if there is a function Ω of z and y, such that XΩ = P (z, y)

∂Ω ∂Ω + Q(z, y) = 0, ∂z ∂y

(z, y) ∈ C[z, y].

(6)

According to the qualitative theory of ordinary differential equations, we learn that although in general we can not explicitly solve system (5), we are occasionally able to find first integrals, that are nonconstant functions p(z, y), analytical on some nonempty open set in C2 and whose derivatives with respect to ζ vanish on the solution curves of equation (4). For some problems, the nontrivial function Ω satisfying two-dimensional autonomous systems under the given parametric conditions enables us to reduce the associated second-order differential equation to a first-order integrable ordinary differential equation. However, how to find the function Ω has been an interesting and challenging problem. In this section, we will use the Divisor Theorem for two variables in the complex domain C to explore the first integral for system (5). In order to make the paper well self-contained and present our results in a straightforward manner, here let us introduce the Divisor Theorem for two variables in the complex domain C: Lemma 1. [Divisor Theorem] Suppose that P0 (ω, z) and Q0 (ω, z) are polynomials in C[ω, z], and P0 (ω, z) is irreducible in C[ω, z]. If Q0 (ω, z) vanishes at all zero points of P0 (ω, z), then there exists a polynomial G(ω, z) in C[ω, z] such that Q0 (ω, z) = P0 (ω, z) · G(ω, z). The proof of the Divisor Theorem can be seen immediately from the following technical theorem [9]: Lemma 2. [Hilbert-Nullstellensatz] Let k be a field and L an algebraic closure of k. Then (i) Every ideal γ of k[X1 , · · · , Xn ] not containing 1 admits at least one zero in Ln . (ii) Let x = (x1 , · · · , xn ), y = (y1 , · · · , yn ) be two elements of Ln . For the set of polynomials of k[X1 , · · · , Xn ] zero at x to be identical with the set of polynomials of k[X1 , · · · , Xn ] zero at y, it is necessary and sufficient that there exists a kautomorphism s of L such that yi = s(xi ) for 1 ≤ i ≤ n. (iii) For an ideal α of k[X1 , · · · , Xn ] to be maximal, it is necessary and sufficient that there exists an x in Ln such that α is the set of polynomials of k[X1 , · · · , Xn ] zero at x. (iv) For a polynomial Q of k[X1 , · · · , Xn ] to be zero on the set of zeros in Ln of an ideal γ of k[X1 , · · · , Xn ], it is necessary and sufficient that there exist an integer m > 0 such that Qm ∈ γ. Now, we are applying the Divisor Theorem to seek the first-integral to system (5). Suppose that z = z(ζ) and y = y(ζ) are the nontrivial solutions of system

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(5), and p(z, y) =

Pm i=0

761

ai (z)y i is an irreducible polynomial in C[z, y] such that p[z(ζ), y(ζ)] =

m X

ai (z)y i = 0,

(7)

i=0

where ai (z) (i = 0, 1, · · · , m) are polynomials of z and all relatively prime in C[z, y], am (z) 6≡ 0. We call formula (7) the first-integral of the polynomial form for system (5). We start our discussions by assuming m = 2 in (7).¯ Note that dp dp ¯ dζ is a polynomial in z and y, and p[z(ζ), y(ζ)] = 0 implies that dζ (5) = 0. By the Divisor Theorem, there exists a polynomial H(z, y) = α(z) + β(z)y in C[z, y] such that the polynomial p(z, y) satisfies ¯ µ ¶¯ dp ¯¯ ∂p ∂z ∂p ∂y ¯¯ = , + dζ ¯(5) ∂z ∂ζ ∂y ∂ζ ¯(5) =

2 X

[a0i (z)y i

· y] +

i=0

"

= [α(z) + β(z)y] ·

2 X

[iai (z)y i−1 (ry + az 2 + bz + d)],

i=0 2 X

# ai (z)y

i

.

(8)

i=0

On equating the coefficients of y i (i=3, 2, 1, 0) on both sides of (8), we have a0 (z) = A(z) · a(z),

(9)

[0, az 2 + bz + d, −α(z)] · a(z) = 0,

(10)

and where a(z) = (a2 (z), a1 (z), a0 (z))t , and  β(z), α(z) − 2r, A(z) =  −2(az 2 + bz + d),

 0, 0 β(z), 0 . α(z) − r, β(z)

Since ai (z) (i = 0, 1, 2) are polynomials, from the first equation a02 (z) = a2 (z)β(z) of (9) we deduce that a2 (z) is a constant and β(z) = 0. For simplification, we take a2 (z) = 1 and solving the linear system (9) gives   1 R . [α(z) − 2r]dz (11) a(z) =  R [a1 (z)α(z) − ra1 (z) − 2(az 2 + bz + d)]dz By (10) and (11), we deduce that deg α(z) = 0 and deg a1 (z) = 1. Assume that a1 (z) = A1 z + A0 , A1 , A0 ∈ C with A1 6= 0. From (11) we have A1 = α(z) − 2r, 3

2 a0 (z) = − 2az 3 − bz +

A1 (A1 +r) 2 z 2

− 2dz + A0 (A1 + r)z + D,

(12)

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where D is an integration constant. Substituting a1 (z) and a0 (z) into (10) and setting the coefficients of z i (i=3, 2, 1, 0) to zero respectively, we obtain an algebraic system A1 a = (− 2a 3 ) · (A1 + 2r) i h A0 a + A1 b = A1 (A21 +r) − b · (A1 + 2r)

(13)

A1 d + A0 b = [(A1 + r)A0 − 2d] · (A1 + 2r) A0 d = D · (A1 + 2r).

Solving system (13) directly, we find that the solutions only exist for the parametric restriction: µ ¶ ¶µ 5 6r2 12r2 2b d= −b − − , (14) 8 25 125a 5a and they are: A1 = −

4r , 5

A0 = −

12r3 2br − , 125a 5a

D=

25 48

µ

6r2 −b 25

¶µ

12r2 2b + 125a 5a

¶2 .

(15)

Substituting a0 (z) and a1 (z) into (7), we derive a first integral of the polynomial form for system (5) as ·

4r 2br y − z+ 5 5a 2

µ

6r2 +1 25b

¶¸

2a 3 z − bz 2 3 µ ¶ 2r2 2 2br2 6r2 − z − 2dz − + 1 z + D = 0. 25 25a 25b

y−

Consequently, a first integral of the polynomial form for equation (4) is obtained as follows · µ ¶¸ 4r 2br 6r2 2a 2r2 2 0 2 (u ) − u+ + 1 u0 − u3 − bu2 − u 5 5a 25b 3 25 µ ¶ 2br2 6r2 −2du − + 1 u + D = 0. (16) 25a 25b It is notable that the first integral (16) of the polynomial form for equation (4) is obtained by applying the Divisor Theorem for two variables in the complex domain C. Thus, under the parametric constraint (14), equation (4) is integrable in Liouville’s sense. If we take a closer look at equation (8), from (12) and (15) we deduce that α(z) = 6r 5 , and thus equation (8) which the polynomial p(z, y) satisfies can be rewritten as 6r dp = p. (17) dζ 5

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Solving equation (17) directly gives us a new quasi-polynomial first integral of an explicit form for equation (4): ½ · ¶¸ µ 4r 2br 6r2 2a 2r2 2 (u0 )2 − u+ + 1 u0 − u3 − bu2 − u 5 5a 25b 3 25 ¶ ¾ µ 6rζ 2br2 6r2 −2du − + 1 u + D e − 5 = I1 , (18) 25a 25b where I1 is an arbitrary integration constant. Note that when d = 0, equation (4) is called the Helmholtz oscillator in the absence of the periodic force, which is a simple nonlinear oscillator with quadratic 2 nonlinearity [4]. From the parametric restraint (14), d = 0 implies b = ± 6r 25 . From formula (18), we can deduce two different first integrals for the Helmholtz oscillator immediately: 2 (I). When b = − 6r 25 , the Helmholtz oscillator has a first integral of the form · ¸ 4ruu0 4r2 u2 2au3 − 6rζ 0 2 (u ) − (19) + − e 5 = I1 . 5 25 3 2

(II). When b = 6r 25 , the Helmholtz oscillator has a first integral of the form · ¸ −6rζ 4r 0 24r3 0 2a 3 8r2 2 24r4 0 2 (u ) − uu − (20) u − u − u − u e 5 = I1 . 5 125a 3 25 625a 2

In the Case (I), i.e. b = − 6r 25 and d = 0, the quasi-polynomial first integral enables us to construct a Hamiltonian function for the Helmholtz oscillator in the absence of the periodic force. The process of construction provides us a motivation of finding traveling wave solutions of closed forms for equation (2) in the next section through using the quasi-polynomial first integral (18), which allows us to reduce the problem of solving equation (4) to that of solving a new first-order ordinary differential equation. We rewrite formula (19) as "µ # ¶2 2ru 2au3 − 6rζ 0 e 5 = I1 . u − − 5 3 Make variable transformations: √ 2rζ η = 2ue− 5 ,

ρ=

√

¶ µ 2ru − 3rζ 0 e 5 , 2 u − 5

then the quasi-polynomial first integral can be re-expressed in terms of ρ, η: √ 2a 3 1 2 ρ − η = I1 (ρ, η). 2 6 Define a function based on the formula I1 (ρ, η): Ã ! √ rζ 1 2 2a 3 H(ρ, η, ζ) = ρ − η e5. 2 6

(21)

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It is easy to check that √ 3rζ = − 2au2 e− 5 = −ρ0ζ , √ ¡ ¢ 2rζ = 2 u0 − 25 ru e− 5 = ηζ0 .

√ 2a 2 rζ 5 2 η e

∂H ∂η

=−

∂H ∂ρ

= ρe

rζ 5

(22)

So, the function H(ρ, η, ζ) is Hamiltonian for the Helmholtz oscillator in the absence of the periodic force. Furthermore, from (22) it is observed that √ ³ rρ ´ rζ 2aη 2rζ rρ rζ 00 0 5 ηζζ = ρζ + e = e 5 + e5, 5 2 5 which can be expressed in terms of u and ζ by making use of variable transformations (21): · ¸ √ − 2rζ 6r2 2e 5 · u00 (ζ) − ru0 (ζ) − au2 + u = 0. 25 2

Therefore, in the case of b = − 6r 25 , H(ρ, η, ζ) is the Hamiltonian function of the Helmholtz oscillator with friction for the integrable case since the solutions to 2 u00 (ζ) − ru0 (ζ) − au2 + 6r 25 u = 0 and the solutions to the Hamilton equations of H(ρ, η, ζ) are the same. It is remarkable that the quasi-polynomial first integral (18) for equation (4) has not been presented previously as far as our knowledge goes. Some analytical results concerning the first integrals of equation (4) or the Helmholtz oscillator in the recent literature are only particular cases of our results (18). For instance, Feng and Huang [17] studied stationary solutions of the KdV-Burgers equation and the integrability of equation (4) by constructing an appropriate integrating factor. Through a compatible condition they established a polynomial first integral which merely coincides with our formula (16). Almendral and Sanjuán [4] investigated the invariance and integrability properties of the Helmholtz oscillator. They used the Lie theory of differential equations to find a first integral under the parametric 2 restraint b = − 6r 25 , which is identical to our formula (19). Recently, Chandrasekar et al. [14] used the modified Prelle–Singer method to construct the first integrals for the Helmholtz oscillator in the absence of the periodic force. Although it was not completely shown in the paper, computations for the solution set (S, R) are complicated, even after a specific assumption of the rational form for S ([14], pp. 2464). Chandrasekar et al. [14] claimed that only under the parametric condition 2 b = − 6r 25 , the Helmholtz oscillator has a first integral of an explicit form as formula (19). Apparently, this statement is incorrect. Moreover, there are several errors on first integrals for the Helmholtz oscillator in ([14], pp. 2464–2465). For example, the solution set (S2 , R2 ) ([14], formula (4.49)) derived by the modified Prelle– Singer method does not satisfy equation (4.46). As a consequence, the second first integral I2 for the Helmholtz oscillator presented in ([14], formula (4.53)) is incorrect either.

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3. Traveling wave solutions In this section, we will use the first integral (18) to derive traveling wave solutions of equation (2). Comparison with the existing results will also be provided at the end of this section. Case 1: when I1 = 0 in formula (18), u0 = du dζ can be expressed in terms of u, i.e., v " ¡ 6r2 ¢ #3 µ 2 ¶ u u 2 2r br 6r t 0 25b + 1 b u = u+ +1 ± au + . (23) 5 5a 25b 3a2 2 r h i Letting k =

6r 2 25b

and making a substitution ψ =

2 3a2

au +

(k+1)b 2

3

, we obtain

an exact solution to equation (4) by solving (23) directly " #2 2r r5 ζ ± 5a e 3a (k + 1) b · . u(ζ) = − r 2 2a e5ζ + c

(24)

where c is an arbitrary integration constant. Changing to the original variables, we obtain a traveling wave solution for equation (2) as · − v (x−vt+ζ0 ) ¸2 6v 2 e 5 (k + 1)α u1 (x, t) = · − v (x−vt+ζ ) . (25) + 0 25β 2β e 5 +c where the velocity v is given by v 4 = 6v 2 (14), and k = − 25α .

625(α2 +4βµ) 36

due to the parametric constraint

Case 2: when I1 < 0, the quasi-polynomial first integral (18) can be expressed as

½· ¸ ¾2 ½· ¸ ¾3 3r 2r 2r br 2 (k + 1) b u0 − u − (k + 1) · e− 5 ζ − 2 au + · e− 5 ζ , 5 5a 3a 2 ½ ·µ ¶ ·µ ¶ ¸¾2 ¸3 d (k + 1)b (k + 1)b 2a − 2r − 2r ζ − 2r ζ ζ 5 5 5 = u+ ·e ·e − u+ ·e , dζ 2a 3 2a ¾2 ½ 2r 2a dφ dq · e− 5 ζ − φ 3 , (26) = dq dζ 3

I1 =

where we assume that

µ ¶ 2r (k + 1)b φ= u+ · e− 5 ζ , 2a

q=

5 rζ e5 . r

The time-dependent first integral (26) becomes an autonomous equation µ ¶2 dφ 2a − φ 3 = I1 . dq 3

(27)

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Equation (27) can be converted to the standard form (φ0 )2 − 4φ3 + g3 = 0 by simply re-scaling it, whose solution can be expressed in terms of the Weierstrass function ℘(q, g2 , g3 ) with invariants g2 = 0 and g3 = −I1 . We know that the Weierstrass function ℘(q, 0, −I1 ) for the standard equation (φ0 )2 − 4φ3 − I1 = 0 can be expressed by the Jacobian elliptic cosine function [1]: ³ p√ √ ´ 1 + cn 2 3Rq + c2 ; 2−4 3 √ ³ p√ φ(q) = R + 3R · √ ´, 1 − cn 2 3Rq + c2 ; 2−4 3 q where R = − 3 I41 and c2 is an arbitrary constant. Consequently, changing to the original variables and using the inverse transformations of φ and q, we obtain a traveling wave solution of equation (2) as ³  v(x−vt+ζ0 ) − 5 + c2 ; 6v 2 c23 − 2v(x−vt+ζ0 )  1 + cn c3 e 5 ³ · u2 (x, t) = e v(x−vt+ζ0 ) 100β 5 1 − cn c3 e− + c2 ;

√ ´ 2− 3 4 √ ´ 2− 3 4

+

 3 + 3 √

(k + 1)α , 2β

where c3 is another arbitrary constant, the velocity v is given by v 4 = 6v 2 and k = − 25α .

(28)

625(α2 +4βµ) 36

(3) When I1 > 0, the standard equation (φ0 )2 − 4φ3 − I1 = 0 has a solution in terms of the Jacobian elliptic cosine function [1]: ³ p√ √ ´ 1 + cn 2 3Rq + c2 ; 2+4 3 √ ³ p√ φ(q) = −R + 3R · √ ´, 1 − cn 2 3Rq + c2 ; 2+4 3 q where R =

3

I1 4 .

Hence, we obtain a traveling wave solution of equation (2) as

³  v(x−vt+ζ0 ) − 5 + c2 ; 6v 2 c23 − 2v(x−vt+ζ0 )  1 + cn c3 e 5 ³ u3 (x, t) = e · v(x−vt+ζ0 ) 100β 5 1 − cn c3 e− + c2 ;

√ ´ 2+ 3 4 √ ´ 2+ 3 4

+ 2

 3 − 3 √

(k + 1)α . 2β

(29)

Note that when d = 0 and b = 6r 25 (i.e. k = 1), the solution formulas (25), (28) and (29) for the Helmholtz oscillator are not presented in the literature, for example, references [4] and [14]. If we take a closer view at the solution formula 2 e2t 2 (25), when c is positive, utilizing the identity 4A[ 1+e 2t ] = −Asech t + 2A tanh t +

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2A, the bounded traveling wave solution (25) can be rewritten as u1 (x, t)

£ v ¤ 3v 2 = − 50β sech2 − 10 (x − vt + ζ0 ) + 2

3v + 25β +

3v 2 25β tanh

(k+1)α 2β ,

£ v ¤ − 10 (x − vt + ζ0 ) (30)

2

where v 4 = 625(α36+4βµ) and ζ0 is arbitrary, which contain the hyperbolic functions secht and tanh t. Since the graph of y = sech2 t looks like a bell-profile solitary wave and the graph of y = tanh t looks like a kink-profile solitary wave (see Figure 2), the solution (30) can be considered as a combination of a bell-profile solitary wave and a kink-profile solitary wave.

Figure 2. The top figure: curve 1 represents the graph of y = sech2 (t/10). The bottom

figure: curve 2 represents the graph of y = tanh(t/10).

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In the pioneering work by Ablowitz and Zeppetella [2], they considered ut = νuxx + su(1 − u).

(31)

Re-scaling equation (31) as t0 = st and x0 = (s/ν)1/2 x and dropping the primes gives ut = uxx + u(1 − u).

(32)

Assume that the traveling wave solution of equation (32) takes the form u(x, t) = u(z) = u(x − ct). Using the Painlevé analysis, Ablowitz and Zeppetella √ reduced equation (32) to the form w00 (z) = 6w2 (z) when the velocity c = ±5/ 6, and obtained that equation (32) has one parameter family of traveling wave solution u(z) =

1 √ , [1 − rexp(z/ 6)]2

(33)

and one more general solution in terms of the Weierstrass function ℘(z, 0, g3 ) ³ (x−ct) ´ 2(x−ct) − √6 − √6 u(x, t) = e ℘ e − k0 , 0, g3 , (34) where k0 and g3 are arbitrary constants. One can see that solution (33) is identical to our formula (25) and solution (34) is in agreement with our results (28) and (29) in the case where k = −1, α = β = 1 and µ = 0. It is remarkable that although some analytical solutions of the Fisher equation (i.e. µ = 0 in equation (2)) in the previous literature have different forms, they are essentially particular cases of our results. For example, Guo and Chen [25] considered the Fisher equation ut = uxx + αu(1 − u),

α > 0.

(35)

Applying the expanded Painlevé analysis, they derived several solution formulas (3.22)-(3.25) for equation (35) ([25], pp. 649). Note that when ε = 1, formula (3.22) is actually equal to (1)

U2 (x, t)

= = =

1

A1 B1 −£ ¤2 , A1 exp[(b/2)η1 ] + B1 exp[(−b/2)η1 ] 1 √ £ ¤2 , 1 + (A1 /B1 )exp[b(x ± 56 6αt)] √ ª2 © exp[−b(x ± 56 6αt)] (36) √ © ª2 , (A1 /B1 ) + exp[−b(x ± 56 6αt)] 1+

A1 B1 exp(bη1 )

where b2 = α/6. It is easy to see that formula (36) is exactly a particular case of our result (25) when c = A1 /B1 and k = −1. For formula (3.24) ([25], pp. 649),

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when ε = 1, it can be rewritten as (3)

A3 B3 +£ ¤2 , 1+ A3 exp[(ib/2)η2 ] + B3 exp[(−ib/2)η2 ] 1 = 1− £ √ ¤2 , 1 + (A3 /B3 )exp[ib(x ± 5i 6αt)] 6 √ © ª2 exp[−ib(x ± 5i 6αt)] 6 = 1− © (37) √ ª2 , (A3 /B3 ) + exp[−ib(x ± 5i 6αt)] 6

U2 (x, t) =

1−

1

A3 B3 exp(ibη2 )

where b2 = α/6. Notice that formula (37) is merely a particular case of our result (25) when c = A3 /B3 and k = 1. For the case of ε = −1, one can easily check in the same manner that formulas (3.22)-(3.25) are particular cases of our result (25) too. It is notable that there is a minor error in formula (3.24) ([25], pp. 649). 2 That is, according to our formula v 4 = 625α in (25), if we take v as a complex 36 √ √ number ± 56 6αi, then we deduce that η2 in formula (3.24) should be x ± 5i 6αt 6 rather than x − εbit. This point can also be verified from (3.15)-(3.18) ([25], pp. 648).

4. Discussion From solution (30), a feature of this solution is a composition of soli³ a bell-profile ´ 2 tary wave and a kink-profile solitary wave. When the velocity v v 4 = 625(α36+4βµ) is a real positive number, the traveling wave solution given √ by (30) has the 2 α2 +4βµ+(k+1)α (k+1)α as ζ = x − vt + ζ0 → +∞ and the limit as limit 2β 2β √ 2 α +4βµ ζ = x − vt + ζ0 → −∞, where k = − . All orders of derivatives of u α with respect to ζ tend to zero as |ζ| → +∞. The amplitudes, wave numbers, and frequencies of traveling wave solutions given by (30) depend on the coefficients α, β and µ. Since the amplitude is inversely proportional to the parameter β, the shock will strengthen if β becomes small. When I1 is not equal to zero, the Weierstrass function ℘ is doubly periodic with an infinite number of poles on the real axis. This implies that all solutions (except (30)) blow up at finite real values of ζ. We know that the Jacobian elliptic cosine function cn(q; m) is a periodic function because of the fact that cn(q + 2K;√m) = −cn(q; m) and cn(q + 4K; m) = cn(q; m), where 2K depends on m = 2−2 3 and is approximately equal to 3.197. Its role is closely similar to π in the trigonometric cosine function. When ´ I1 is negative, we consider the case of ³ 625(α2 +4βµ) 4 and ζ = x − vt + ζ0 → +∞ for solution the positive velocity v v = 36 v(x−vt+ζ0 )

5 formula (28), then we deduce that q1 = c3 e− → 0 as ζ → +∞. Taking c2 = 4nK with n ∈ Z in (28) and using the identity cn(q; m) = 1 − 12 q 2 + o(q 4 )

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[1], we obtain the limit of solution (28) as ζ → +∞ ³ √ ´  √ 2− 3 1 + cn q ; 1 4 3  (k + 1)α 6v 2  ³ q1 · , lim u2 (x, t) = lim + √ ´ + q1 →0 100β ζ→+∞ 3 2β 1 − cn q1 ; 2−4 3 " √ # 6v 2 2 4 − q12 3 (k + 1)α = lim q1 · + + , 2 q1 →0 100β q1 3 2β p 2 α2 + 4βµ + (k + 1)α = . 2β 

2

When I1 is positive, the same limit can also be derived for the case of the positive velocity√v and ζ = x − vt + ζ0 → +∞, but the only difference lies in that when m = 2+2 3 , cn(q; m) is a periodic function with 2K 0 ≈ 5.535.

5. Conclusion As we know, traveling wave solutions are solutions of special type, which can be usually characterized as solutions invariant with respect to translation in space. From the physical point of view, traveling waves usually describe transition processes. Transition from one equilibrium to another is a typical case, although more complicated situations can arise. These transition processes usually “forget” their initial conditions and reflect the properties of the medium itself. The theory of traveling wave solutions of reaction-diffusion equations is commonly considered as one of the fast developing areas of modern mathematics. The history of this theory dates back to the famous mathematical work by Luther [33], and Kolmogorov, Petrovsky and Piscunov [31] and with works in chemical physics. In the past century, many ingenious techniques have been proposed for obtaining traveling wave solutions of differential equations [3, 8, 26, 28, 29, 40] etc. In this work, we proposed an effective method to deal with traveling wave solutions of a reaction-diffusion equation based on the ring theory of commutative algebra. Applying the Divisor theorem we obtained a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then using this first integral we reduce the reaction-diffusion equation (2) to a first-order integrable ordinary differential equation. A class of traveling wave solution was accordingly derived by solving this first-order differential equation. Comparisons with the existing results in the literature were provided, which indicates that some analytical results in the literature contain errors. The technique described herein can be applied to other equations of biological and physical interest.

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Acknowledgments Main contents of this paper were presented at the SIAM Conference on Analysis of Partial Differential Equations (PD07), Phoenix, Arizona, December 10–12, 2007 (http://www.siam.org/meetings/pd07/index.php). The work is supported by NSF(USA) Grant CCF-0514768, and partially supported by UTPA Faculty Research Council Grant 119100.

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Shenzhou Zheng Department of Mathematics Beijing Jiaotong University Beijing, 100044 China e-mail: [email protected] David Y. Gao Department of Mathematics Virginia Tech. University Blacksburg, VA 24061 USA e-mail: [email protected] (Received: August 18, 2008) Published Online First: March 4, 2009

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