Exact Traveling Wave Solutions for the Modified

Exact Traveling Wave Solutions for the Modified Kawahara Equation Ahmed Elgarayhi Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Reprint requests to Prof. A. E.; King Saud University, Al-Qassem Branch, College of Science, P.O. Box 237, Bureidah 81999, KSA. E-mail: [email protected] Z. Naturforsch. 60a, 139 – 144 (2005); received October 21, 2004 The mapping method is used with the aid of the symbolic computation system Mathematica for constructing exact solutions of the modified Kawahara equation. By this method the modified Kawahara equation is investigated and new exact traveling wave solutions are obtained. The solutions obtained in this paper include Jacobi elliptic solutions, combined Jacobi elliptic solutions, solitary wave solutions, periodic wave solutions, trigonometric solutions and rational solutions. Key words: Modified Kawahara Equation; Periodic Wave Solution; Mapping Method; Jacobi Elliptic Functions.

1. Introduction In the last decades, direct search for exact solutions of nonlinear partial differential equations (NLPDEs) has become increasingly attractive partly due to the availability of computer symbolic software like Mathematica or Maple, which allows to perform complicated and tedious algebraic calculations and helps to find exact solutions of NLPDEs [1 – 4]. There has been much activity aiming at finding powerful methods for obtaining such solutions. We can cite the Backlund transformation [5], the Darboux transformation [6], the Jacobi elliptic function method [7], the tanh-function method [8], the sine-cosine function method [9, 10], the homogenous balance method [11], and the Jacobi function expansion method [12, 13]. Very recently, a unified method called the mapping method has been developed to obtain Jacobi elliptic functions, solitons and periodic solutions to some NLPDEs [14 – 16]. A remarkable observation about the mapping method is that it allows to find both Jacobi elliptic functions, triangular functions and solitons using the same procedure. Moreover, this method permits the classification of solutions depending on four parameters. The motivation of this paper is to use the mapping method for constructing new traveling wave solutions to the modified Kawahara equation. 2. Elliptic Equation and its Solutions The main idea of the mapping method is to take full advantage of the elliptic equation that Jacobi elliptic

functions satisfy and use its solutions to obtain new periodic and solitonic solutions of the modified Kawahara equation. The basic idea is as follows. For a given nonlinear partial differential equation N(u, ut , ux , uxx , . . .) = 0,

(1)

we seek its traveling wave solution of the form u(x,t) = u(ζ ),

ζ = kx − λ t.

(2)

Substituting (2) into (1) yields an ordinary differential equation of u(ζ ). Then u(ζ ) is expanded into a polynomial in f (ζ ), n

u(ζ ) = ∑ ai f i ,

(3)

i=0

where ai are constants to be determined and n is fixed by balancing the linear term of the highest order derivative with the nonlinear term. In the present work, we shall introduce the following new auxiliary ordinary differential equation f (ζ ) = p f 2 (ζ ) + q f 4 (ζ ) + s f 6 (ζ ) + r. (4) Here the prime means derivatives with respect to ζ and p, q, s, and r are constants. After (3) with (4) is substituted into the ordinary differential equation, the coefficients ai , k, λ , p, q, s, and r may be determined. Thus (3) establishes an algebraic mapping relation between the solution of (1) and that of (4). We shall construct exact solutions for the modified Kawahara equation by using the solutions of (4) shown in Table 1.

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A. Elgarayhi · Exact Traveling Wave Solutions for the Modified Kawahara Equation

140 Case 1

Arbitrary constants p = −(1 + m2 ), q = 2m2 , s = 0, r = 1

Solutions of (1) f (ζ ) = snζ , f (ζ ) = cdζ

2

p = 2m2 − 1, q = −2m2 , s = 0, r = 1 − m2

f (ζ ) = cnζ

3

p = 2 − m2 , q = −2, s = 0, r = m2 − 1

f (ζ ) = dnζ

4

p = 2m2 − 1, q = 2, s = 0, r = −m2 (1 − m2 )

f (ζ ) = dsζ

5

p = 2 − m2 − 1, q = 2, s = 0, r = 1 − m2

f (ζ ) = csζ

6

p=

7

p=

m2 −2 2 , m2 −2 2 ,

q= q=

m2 2 , m2 2 ,

s = 0, r = s = 0, r =

f (ζ ) =

1 4 m2 4

snζ 1±dnζ

f (ζ ) = snζ ± icnζ , i2 = −1, f (ζ ) = √

dnζ

f (ζ ) =

√

i

p=

8

1−2m2 2 ,

q = 12 , s = 0, r =

Table 1. Different solutions of the elliptic equation (4).

1 4

f (ζ ) = f (ζ ) =

1−m2 snζ ±dnζ dnζ

mcnζ ±i 1−m2 √ cnζ

,

1−m2 snζ ±dnζ snζ 1±cnζ

,

f (ζ ) = msnζ ± idnζ 9

p=

10

p=

m2 +1 2 , 1+m2 2 ,

p=

1+m2 2 ,

12

p=

13

p=

1+m2 2 , m2 −2 2 ,

14

p = 0, q = 2, s = 0, r = 0

11

m2 −1 4 2 = s = 0, r = 1−m 4 (1−m2 )2 = − 12 , s = 0, r = − 4 2 )2 = (1−m , s = 0, r = 14 2 2 = m2 , s = 0, r = 14

q= q q q q

m2 −1 2 , 1−m2 2 ,

f (ζ ) =

s = 0, r =

f (ζ ) =

dnζ 1±msnζ cnζ 1±snζ

f (ζ ) = mcnζ ± dnζ f (ζ ) = f (ζ ) =

snζ dnζ ±cnζ ζ √ cn 1−m2 ±dnζ

f (ζ ) = C/ζ

Table 2. Jacobi elliptic functions degenerate into hyperbolic functions when the modulus is approaching 1.

Table 3. Jacobi elliptic functions degenerate into trigonometric functions when the modulus is approaching 0.

snζ → tanh ζ sdζ → sinh ζ ndζ → cosh ζ

snζ → sin ζ sdζ → sin ζ ndζ → 1

cnζ → sechζ cdζ → 1 csζ → cschζ

dnζ → sechζ dcζ → 1 dsζ → cschζ

scζ → sinh ζ nsζ → coth ζ ncζ → cosh ζ

Here C is a constant. The Jacobi elliptic functions sn(ζ |m), cn(ζ |m), dn(ζ |m), where m (0 < m < 1) is the modulus of the elliptic function, are doubly periodic and possess properties of triangular functions. In addition we see that other solutions are obtained from Table 1 in case of degeneracy. As we know, when m → 1, Jacobi elliptic functions degenerate as hyperbolic functions as indicated in Table 2. When m → 0, Jacobi elliptic functions degenerate into trigonometric functions as shown in Table 3. 3. The Modified Kawahara Equation We consider the modified Kawahara equation ut + ux + u2ux + α uxxx + β uxxxxx = 0,

(5)

where α and β are nonzero real constants. This equation arises in the theory of shallow water waves [17], and its exact solutions were obtained by using the tanh-function method [18] and the sech-function

cnζ → cos ζ cdζ → cos ζ csζ → cot ζ

dnζ → 1 nsζ → csc ζ dsζ → csc ζ

scζ → tan ζ ncζ → sec ζ dcζ → sec ζ

method [19]. After using the transformation u(x,t) = u(ζ ), ζ = x − λ t, and integrating once, (5) becomes 1 (1 − λ )u + u3 + α u + β u(4) = 0. 3

(6)

The solution of (6) may be chosen as n

u(ζ ) = ∑ ai f i (ζ ),

(7)

i=0

with arbitrary constants a i (i = 0, 1, . . . n) to be determined later. Balancing the highest derivative term u (4) with the highest power nonlinear term u 3 gives the leading order n = 2. Substituting (7) into (6) along with (4) and using Mathematica yields a system of equations with respect to f i (ζ ). Setting the coefficients of f i (ζ ) in the obtained system of equations to zero, we get the following set of algebraic equations a30 + 6a2r(α + 4pβ ) − 3a0(λ − 1) − 3c1 = 0,


3a1 (1 + a20 + pα + p2 β + 6qrβ − λ ) = 0,

141

Case 2

a0 a21 + a20 a2 + a2 (1+ 4pα + 16p 2β + 36qrβ − λ ) = 0, a1 (a21 + 6a0a2 + 3qα + 30pqβ + 60rsβ ) = 0, 3a2 (a21 + a0 a2 + 3qα + 60pqβ + 80rsβ ) = 0, 3a1 [a22 + 6q2β + s(α + 26pβ )] = 0, a2 {a22 + 2[45q2β + 4s(α + 40pβ )]} = 0, 60sβ a1 q = 0, 240sβ a2q = 0,

15c1 β [α +20(−1+2m2)β ] 900c1 m2 β 2 2 − cn (ζ ), A2 A2 α2 λ = 1− − 72m2(1 − m2)β − 24(−1 + 2m2)2 β , 10β

u=

A2 1 √ − 23 , β 15 10 3 A2 = α − 120[6m2(1 − m2) + 2(−1 + 2m2)2 ]αβ 2 +800(−1 + 2m 2)[18m2 (1 − m2) + 4(−1 + 2m2)2 ]β 3 . c1 =

35s2 β a1 = 0, 128s2 a2 β = 0.

(8)

Solving the above system, we get 1. a0 =

Case 3 15c1 β [α + 20(2 − m2)β ] 900c1 β 2 2 − dn (ζ ), A3 A3 α2 λ = 1− − 24(2 − m2)2 β − 72(−1 + m2)β , 10β

u=

15c1 β (α +20pβ ) , α 3 −120(2p2 −3qr)αβ 2 +800p(4p2 −9qr)β 3

a1 = 0, a2 =

450c1 qβ 2 , α 3 −120(2p2 −3qr)αβ 2 +800p(4p2 −9qr)β 3

c1 =

1 √ 15 10

3 2 2 2 3 2 − (α −120(2p −3qr)αββ 3+800p(4p −9qr)β ) ,

α 2 λ = 1 − 10 β − 24p β + 36qr β . 2

(9a)

A2 1 √ − 33 , β 15 10 3 A3 = α − 120[2(2 − m 2)2 + 6(−1 + m2)]αβ 2 + 800(2 − m2)[4(2 − m2)2 + 18(−1 + m2)]β 3 . c1 =

2. 3(3c1 p4 − 2c1 p2 qr) , 6 (171p − 294p4qr − 204p2q2 r2 − 40q3r3 )β 3 a1 = 0,

a0 = a2 =

18c1 p3 q , (171p6 −294p4 qr−204p2 q2 r2 −40q3 r3 )β 3

c1 =

1 3

6 4 2 q2 r2 −40q3 r3 )2 β 3 5 − (−171p −294p qr−204p , 2 p6

2λ = 2 − 53p2β − +52qrβ − 20qp2r β , 2 2

α = −5p − 10qr p .

(9b)

Substituting (9a) into (7) and using the solution of (4), we obtain: Case 1 u=

15c1

β [α + 20(−1 − m2)β ]

+

m2 β 2

900c1 A1

A1 α2 λ = 1− + 72m2β − 24(−1 − m2)2 β , 10β

sn2 (ζ ),

A2 1 √ − 13 , β 15 10 3 A1 = α − 120[−6m2 + 2(−1 − m2)2 ]αβ 2

c1 =

+ 800(−1 − m2)[−18m2 + 4(−1 − m2)2 ]β 3 .

Case 4 15c1β [α + 20(−1 + 2m2)β ] 900c1β 2 2 + ds (ζ ), A4 A4 α2 λ = 1− − 72m2(1 − m2)β − 24(−1 + 2m2)2 β , 10β u=

A2 1 √ − 43 , β 15 10 3 A4 = α − 120[6m2(1 − m2) + 2(−1 + 2m2)2 ]αβ 2 c1 =

+800(−1 + 2m 2)[18m2 (1 − m2) + 4(−1 + 2m2)2 ]β 3 .

Case 5 15c1 β [α + 20(2 − m2)β ] 900c1 β 2 2 − cs (ζ ), A5 A5 α2 λ = 1− + 72(1 − m2)β − 24(2 − m2)2 β , 10β u=

A2 1 √ − 53 , β 15 10 3 A5 = α − 120[−6(1 − m 2) + 2(2 − m2)2 ]αβ 2

c1 =

+ 800(2 − m2)[−18(1 − m2) + 4(2 − m2)2 ]β 3 .


142

Case 6 u=

15c1 β [α + 20(2 − m2)β ] 900c1β 2 sn2 (ζ ) − , A6 A6 (1 ± dn(ζ ))2

A2 α2 9m2 β 1 − 6(−2 + m2)2 β , c1 = √ λ = 1− + − 63 , 10β 2 β 15 10 2 1 3m 9m2 + (−2 + m2)2 αβ 2 + 400(−2 + m2) − + (−2 + m2)2 β 3 . A6 = α 3 − 120 − 8 2 8 Case 7 u=

15c1 β [α + 10(−2 + m2)β ] 225c1m2 β 2 + (snζ ± i cnζ )2 A7 A7

A2 α2 9m4 β 1 − 6(−2 + m2)2 β , c1 = √ + − 73 , 10β 2 β 15 10 4 1 3m 9m4 3 2 2 2 2 2 2 + (−2 + m ) αβ + 400(−2 + m ) − + (−2 + m ) β 3 . A7 = α − 120 − 8 2 8

λ = 1−

Case 8 15c1 β [α + 10(1 − m2)β ] 225c1β 2 sn2 ζ + , A8 A8 (1 ± cnζ )2 A2 α2 9β 1 2 2 − 6(1 − m ) β , c1 = √ λ = 1− + − 83 , 10β 2 β 15 10 1 3 9 A8 = α 3 − 120 − + (1 − m2 )2 αβ 2 + 400(1 − m2) − + (1 − m2)2 β 3 . 8 2 8 u=

Case 9 u=

dn2 ζ 15c1 β [α + 10(1 + m2)β ] 225c1(−1 + m2)β 2 + , A9 A9 (1 ± msnζ )2

A2 α2 9 1 + (−1 + m2)2 β − 6(1 + m2)2 β , c1 = √ − 93 , 10β 2 β 15 10 3 1 9 3 2 2 2 2 2 2 2 2 2 2 A9 = α − 120 − (−1 + m ) + (1 − m ) αβ + 400(1 + m ) − (−1 + m ) + (1 + m ) β 3 . 8 2 8

λ = 1−

Case 10 u=

15c1 β [α + 10(1 + m2)β ] 225c1 (1 − m2)β 2 cn2 ζ + , A10 A10 (1 ± snζ )2

A2 α2 9 1 2 2 2 2 λ = 1− + (1 − m )(−1 + m )β − 6(1 + m ) β , c1 = √ − 10 , 10β 2 β3 15 10 3 1 9 A10 = α 3 −120 − (1−m2 )(−1+m2 )+ (1 + m2)2 αβ 2 +400(1+m2) − (1−m2 )(−1+m2 )+(1+m2 )2 β 3 . 8 2 8 Case 11 u=

15c1 β [α + 10(1 + m2)β ] 225c1β 2 − (mcnζ ± dnζ )2 , A11 A11

A2 α2 9 1 λ = 1− − (−1 + m2)β − 6(1 + m2)2 β , c1 = √ − 11 , 10β 2 β3 15 10 3 9 1 3 2 2 2 2 2 2 2 2 A11 = α − 120 (−1 + m ) + (1 + m ) αβ + 400(1 + m ) (−1 + m ) + (1 + m ) β 3 . 8 2 8


143

Case 12 u=

sn2 ζ 15c1 β [α + 10(1 + m2)β ] 225c1(1 − m2)2 β 2 + , A12 A12 (dnζ ± cn ζ )2

A2 α2 9 1 + (1 − m2)2 β − 6(1 + m2)2 β , c1 = √ − 12 , 10β 2 β3 15 10 3 1 9 A12 = α 3 − 120 − (1 − m2)2 + (1 + m2)2 αβ 2 + 400(1 + m2) − (1 − m2)2 + (1 + m2)2 β 3 . 8 2 8

λ = 1−

Case 13 cn2 ζ 15c1 β [α + 10(−2 + m2)β ] 225c1m2 β 2 + √ 2 , A13 A13 1 − m2 ± dnζ A2 α2 9m2 β 1 λ = 1− + β − 6(−2 + m2)2 β , c1 = √ − 13 , 10β 2 β3 15 10 3m2 1 9m2 + (−2 + m2)2 αβ 2 + 400(−2 + m2) − + (−2 + m2)2 β 3 . A13 = α 3 − 120 − 8 2 8 u=

Case 14

spectively:

15c1 β 900c1β 2 C 2 + , u= α2 α3 ζ

λ = 1−

α2 , 10β

c1 =

1 √ 15 10

−

15c1β (α − 40β ) α 3 − 240αβ 2 + 3200β 3 900c1β 2 + 3 tanh2 (ζ ), α − 240αβ 2 + 3200β 3 α2 λ = 1− − 24β , 10β

u=

α6 . β3

In case of degeneracy, and due to the large number of solutions in Table 1, it is not advisable to treat every case. Therefore we shall only deal with a few cases as illustrative examples. If m → 1, we can obtain the following soliton solutions from the Cases 1 and 2, re-

c1 =

1 √ 15 10

−

(10)

(α 3 − 240αβ 2 + 3200β 3)2 , β3

15c1β (α + 20β ) α 3 − 240αβ 2 + 3200β 3 900c1β 2 − 3 sech2 (ζ ), α − 240αβ 2 + 3200β 3 α2 λ = 1− − 24β , 10β u=

c1 =

1 √ 15 10

−

(11)

(α 3 − 240αβ 2 + 3200β 3)2 . β3

This solution is satisfactory when compared with that found previously by Sirendaoreji [19], see Figures 1 and 2. Fig. 1. A solitary wave solution of (11) is shown at α = 10, β = −5.

If m → 0, we can find the following trigonometric solutions from Cases 4 and 5, respectively:


144

λ = 1−

α2 − 24β , 10β

1 c1 = √ 15 10

(α 3 − 240αβ 2 − 3200β 3)2 − . β3

(13)

I point out that all above solutions, except those from (10) to solution (13), are new and have not been reported in the literature. Moreover, only some solutions of (5) are shown.

4. Conclusion Fig. 2. A solitary wave solution by Sirendaoreji [19].

u=

15c1 β (α − 20β ) 3 α − 240αβ 2 − 3200β 3 900c1 β 2 + 3 α − 240αβ 2 − 3200β 3 α2

csc2 (ζ ),

λ = 1−

− 24β , 10β 1 (α 3 − 240αβ 2 − 3200β 3)2 c1 = √ − , β3 15 10 u=

(12)

15c1 β (α + 40β ) α 3 − 240αβ 2 − 3200β 3 +

900c1β 2 cot2 (ζ ), α 3 − 240αβ 2 − 3200β 3

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Exact solutions of the modified Kawahara equation are studied by the mapping method. In this paper, we construct explicit and new solutions such as Jacobi elliptic solutions, combined Jacobi elliptic solutions, solitary wave solutions, periodic wave solutions and shock wave solutions, trigonometric solutions and rational solutions. The idea of our method is to use the elliptic equation involving four parameters instead of specific functions in previous methods [5 – 13]. Therefore, many tedious and repetitive calculations can be avoided. Moreover, this method can be applicable to a large variety of nonlinear partial differential equations. Acknowledgements I would like to express my sincere thanks to the referees for their useful comments.

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