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A rational generalization of Fan’s method and its application to generalized shallow water wave equations
Applied Mathematics and Computation 216 (2010) 1984–1995
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A rational generalization of Fan’s method and its application to generalized shallow water wave equations Zonghang Yang a,*, Y.C. Hon b a b
Department of Information Systems, City University of Hong Kong, Hong Kong SAR, PR China Department of Mathematics, City University of Hong Kong, Hong Kong SAR, PR China
a r t i c l e
i n f o
a b s t r a c t In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations. Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved.
Keywords: The GSWW equation Fan’s method Rational expansion Travelling wave solution
1. Introduction The Boussinesq approximation theory [1] for generalized classical shallow water wave leads to the GSWW equation:
v xxxt þ av x v xt þ bv t v xx v xt v xx ¼ 0; where a and b are non-zero constants. Using
uxxt þ auut
bux @ 1 x ut
ð1:1Þ
v x ¼ u in [2], Eq. (1.1) can be simplified to
ut ux ¼ 0;
ð1:2Þ
which has recently attracted many attentions from researchers under the following cases: (1) For the case when a ¼ 2b, the following AKNS–SWW equation
uxxt þ 2buut bux @ 1 x ut ut ux ¼ 0;
ð1:3Þ
was discussed by Ablowitz et al. [3]. (2) For the case when a ¼ b, the following equation
uxxt þ buut bux @ 1 x ut ut ux ¼ 0;
ð1:4Þ
was discussed by Hirota and Satsuma [4]. The GSWW Equation (1.1) in potential form was studied by Clarkson and Mansfield [2,5] who gave a complete catalog of classical and nonclassical symmetry reductions. The necessary conditions of Painlevé tests was given by Ablowitz et al. [6] and the complete integrability of Eq. (1.1) for the case when a ¼ 2b or a ¼ b was established by Weiss et al. [7]. It had been proven by Hietarinta [8] that the GSWW Equation (1.2) can be expressed in Hirota’s bilinear form and when a ¼ 2b, Eq. (1.2)
* Corresponding author. E-mail addresses: [email protected], [email protected] (Z. Yang), [email protected] (Y.C. Hon). 0096-3003/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.029
Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
1985
can be reduced to (1.3) or to (1.4) when a ¼ b. In other words, both Eqs. (1.3) and (1.4) are solvable by using Hirota’s bilinear method [9]. The N-soliton solutions for Eqs. (1.3) and (1.4) had also been found using this technique [4]. Although there is no scaling transformation that can reduce (1.3) to (1.4), the classical methods of Lie, the nonclassical method of Bluman and Cole, and the direct method of Clarkson and Kruskal can be applied to solve the GSWW Equation (1.2) to obtain some kinds of symmetry reductions [10,11]. Recently, Elwakil found a lot of exact solutions by using the modified extended tanh-function method [12]. Since nonlinear PDEs are widely used to describe complex phenomena in various field of sciences, various methods for obtaining exact travelling solitary wave solutions to nonlinear evolution equations have been proposed. For instances, inverse scattering method [13,14], Darboux transformation [15–20], Hirota bilinear method [21,22], Painleve’ expansions [23], solitary wave ansatze method for Boussinesq equation [24,25], Homogeneous balance method [26] and tanh-function method [27–34]. Among these methods, the tanh-function method provides a straightforward and effective algorithm to obtain particular solutions for system of nonlinear partial differential equations by taking the advantage that solitary solutions are essentially of localized nature. Based on the solution to the well-known Riccati equation, Fan presented a useful extended tanh-function method [31] and further extended this method to solve many different kinds of partial differential equations [32]. Due to the importance of Fan’s generalization, this method is also named as Fan’s method. Recently, Elwakil presented a modified extended tanh-function method [33] and Zheng, Chen and Zhang developed a generalized extended tanh-function method [34]. In this paper we devised a rational generalization for Fan’s method and successfully solved GSWW Eq. (1.2). To the acknowledge of the authors, some of the obtained solutions have never been given before. 2. Methodology To illustrate the idea of our proposed method, we first consider the following ordinary differential equation (ODE)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x0 ¼ e c0 þ c1 x þ c2 x2 þ c3 x3 þ c4 x4 ;
ð2:1Þ
where x ¼ xðnÞ, and e ¼ 1, 0 denotes d=dn. For different choices of c0 ; c1 ; c2 ; c3 and c4 , Eq. (2.1) admits different kinds of fundamental solutions which can be used to construct the exact solution of nonlinear PDEs. The details of the solution of Eq. (2.1) are presented in Appendix A. Consider the following nonlinear partial differential equation
Hðu; ut ; ux ; uxx ; . . .Þ ¼ 0:
ð2:2Þ
The main steps of our proposed method are outlined as follows. Step 1. By using the wave transformation uðx; tÞ ¼ UðnÞ; n ¼ x þ ct (c is a non-zero constant), we reduce Eq. (2.2) into the following ODE
HðU; U 0 ; U 00 ; . . .Þ ¼ 0;
ð2:3Þ
0
where U ¼ dU=dn. Step 2. Introduce a new variable x ¼ xðnÞ which is a solution of the following ODE
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u r x ¼ et cj xj ; 0
ð2:4Þ
j¼0
where cj ; ðj ¼ 0; . . . ; rÞ are constants to be determined. The high order derivatives with respect to the variable n become:
x00 ¼
r eX
2
jcj xj1 ; . . .
ð2:5Þ
j¼1
Step 3. By virtue of the new variable x, we expand the solution of Eq. (2.3) as
uðx; tÞ ¼ UðnÞ ¼
n X
ai xi þ
i¼0
n X
bi xi ;
ð2:6Þ
i¼1
where ai ði ¼ 0; 1; . . . ; nÞ; bi ði ¼ 1; 2; . . . ; nÞ are constants to be determined. This expansion develops the polynomial expansion in Fan’s method [31,32] as
uðx; tÞ ¼ UðnÞ ¼
n X i¼0
ai xi ;
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by adding rational terms. To determine the values of n and r, we substitute (2.4) and (2.5) into (2.3) and balance the highest order linear term with the nonlinear terms in (2.3) to obtain a relation for n and r. For illustration, in solving the following Burgers’ equation
ut þ uux uxx ¼ 0; balancing the terms uux and uxx we obtain
1 n þ n 1 þ r ¼ n 2 þ r; 2
ð2:7Þ
and hence 2n ¼ r 2. The values of n and r then lead to the series expansion of the travelling wave solutions for (2.2). For instance, if we take n ¼ 1 and r ¼ 4 in (2.7), we can use the following expansion as a solution of the Burgers’ equation
u ¼ a0 þ a1 x þ b1 x1 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x0 ¼ e c0 þ c1 x þ c2 x2 þ c3 x3 þ c4 x4 : If we take n ¼ 2 and r ¼ 6 in (2.7), we have
u ¼ a0 þ a1 x þ a2 x þ b1 x1 þ b2 x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x0 ¼ e c0 þ c1 x þ c2 x2 þ c3 x3 þ c4 x4 þ c5 x5 þ c6 x6 : We can then take r ¼ 4 and make use of the fundamental solutions of the ODE to derive the solutions of (2.2). Step 4. Substituting the expansions (2.4) and (2.5) into Eq. (2.3) and setting the coefficients of all powers of xi and qP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r i j x j¼0 cj x to zero, we obtain a system of algebraic equations, from which the constants ai ; ði ¼ 0; 1; . . . ; nÞ; bi ; ði ¼ 1; 2; . . . ; nÞ; cj ; ðj ¼ 0; 1; . . . ; rÞ can be found explicitly by using symbolic software such as mathematica. Step 5. Analyzing the parameters cj ; ðj ¼ 0; 1; . . . ; rÞ obtained in Step 4, we can then construct many kinds of solutions if (2.3) is of exact solvability. In this paper we only consider the case when r ¼ 4 due to the efficiency in using the fundamental solutions. It is noted here that the solution of the system of algebraic equations will become tedious with an increase of the values of n and r. Substituting ai ; ði ¼ 0; 1; . . . ; nÞ; bi ; ði ¼ 1; 2; . . . ; nÞ and the fundamental solutions of Eq. (2.3) into (2.6), we can now obtain the exact travelling solutions for Eq. (2.2). 3. Exact solutions for the GSWW equation Using the wave transformations u ¼ UðnÞ; n ¼ x þ ct, we transform Eq. (1.2) to the following ODE
cU 000 þ cða þ bÞUU 0 ðc þ 1ÞU ¼ 0:
ð3:1Þ
From (2.6) we expand the solution of (3.1) as
U¼
n X
a i xi þ
i¼0
n X
bi xi ;
ð3:2Þ
i¼1
where x satisfies (2.4). Balancing the highest order linear terms with the nonlinear terms in (3.1) gives
3 1 n 3 þ r ¼ n þ n 1 þ r: 2 2
ð3:3Þ
From (3.3) we get the relation n ¼ r 2 which we choose n ¼ 2 and r ¼ 4 to obtain
U ¼ a0 þ a1 x þ a2 x2 þ b1 x1 þ b2 x2 ;
ð3:4Þ i
where x satisfies (2.1). Substituting (3.4), (2.4) and (2.5) into (3.1) and setting the coefficients of all powers like x and qP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r j xi j¼0 cj x to zero, we then obtain a system of algebraic equations with the aid of Mathematica. The system of algebraic equations is given as follows 2
2
2 c a e b2 þ 2 c b e b2 þ 24 c e3 b2 c0 ¼ 0; 3 c a e b1 b2 þ 3 c b e b1 b2 þ 6 c e3 b1 c0 þ 15 c e3 b2 c1 ¼ 0; 2
2
c a e b1 þ c b e b1 2 e b2 2 c e b2 þ 2 c a e a0 b2 þ 2 c b e a0 b2 þ 3 c e3 b1 c1 þ 8 c e3 b2 c2 ¼ 0; e b1 c e b1 þ c a e a0 b1 þ c b e a0 b1 þ c a e a1 b2 þ c b e a1 b2 þ c e3 b1 c2 þ 3 c e3 b2 c3 ¼ 0;
e a1 þ c e a1 c a e a0 a1 c b e a0 a1 c a e a2 b1 c b e a2 b1 3 c e3 a2 c1 c e3 a1 c2 ¼ 0; c a e a21 c b e a21 þ 2 e a2 þ 2 c e a2 2 c a e a0 a2 2 c b e a0 a2 8 c e3 a2 c2 3 c e3 a1 c3 ¼ 0; 3 c a e a1 a2 3 c b e a1 a2 15 c e3 a2 c3 6 c e3 a1 c4 ¼ 0; 2 c a e a22 2 c b e a22 24 c e3 a2 c4 ¼ 0:
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Solving the above algebraic system with the aid of Mathematica, we list the solutions of the system under the following several cases: Case 1. c0 ¼ c1 ¼ c3 ¼ 0; c4 – 0
1 ; c2 1 a ¼ b; a1 ¼ 0; a2 ¼ 0; c ¼ 1; c2 ¼ 0; 1 þ c 4 c c2 12 c4 a1 ¼ 0; b1 ¼ 0; b2 ¼ 0; a0 ¼ ; a2 ¼ : ca þ cb aþb
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 0; c ¼
From (A.1), (A.2), (A.3) and (3.4), we find a hyperbolic type solution, a triangular type solution and two polynomial type solutions when a þ b ¼ 0, a bell shaped solitary wave solution, a triangular type solution and a rational type solution when a þ b – 0:
rffiffiffiffiffiffiffiffiffi pffiffiffiffiffi c4 1 cosh c2 x þ t c2 > 0; c4 < 0; c2 1 c2 rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi c4 1 c2 x þ ¼ a0 þ b1 cos t c2 < 0; c4 > 0; c2 1 c2 pffiffiffiffiffi b1 c 4 ¼ a0 ðx tÞ c2 ¼ 0; c4 > 0; e ffiffiffiffiffi p b1 c4 ðx tÞ ¼ a0 þ þ b2 c4 ðx tÞ2 c2 ¼ 0; c4 > 0;
u11 ¼ a0 þ b1 u12 u13 u14
e
when a þ b ¼ 0, and
1 þ c 4cc2 12c2 2 pffiffiffiffiffi þ sech ð c2 ðx þ ctÞÞ c2 > 0; c4 < 0; ca þ cb aþb pffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c2 ¼ þ sec2 ð c2 ðx þ ctÞÞ c2 < 0; c4 > 0; ca þ cb aþb 1þc 12 ¼ c2 ¼ 0; c4 > 0; ca þ cb ða þ bÞðx þ ctÞ2
u15 ¼ u16 u17
when a þ b – 0. Case 2. c1 ¼ c3 ¼ 0
a1 ¼ 0;
b1 ¼ 0;
a1 ¼ 0;
a2 ¼ 0;
a1 ¼ 0;
b1 ¼ 0;
1 þ c 4 c c2 12 c4 ; a2 ¼ ; ca þ cb aþb 1 þ c 4 c c2 12 c0 b1 ¼ 0; a0 ¼ ; b2 ¼ ; ca þ cb aþb 1 þ c 4 cc2 12 c4 12 c0 a0 ¼ ; a2 ¼ ; b2 ¼ : ca þ cb aþb aþb b2 ¼ 0;
a0 ¼
If c0 ; c2 ; c4 satisfy c0 ¼ c22 =4c4 , from (A.4), (A.5) and (3.4), we find three solitary wave solutions and three triangular type solutions when a þ b – 0:
u21 u22 u23 u24 u25 u26
rffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 6c2 c2 2 ðx þ ctÞ ; c2 < 0; c4 > 0; ¼ þ tanh ca þ cb aþb 2 rffiffiffiffiffi 1 þ c 4cc2 6c2 c 2 ðx þ ctÞ ; c2 > 0; c4 < 0; ¼ tan2 ca þ cb aþb 2 rffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 6c2 c2 2 ¼ ðx þ ctÞ ; c2 < 0; c4 > 0; þ coth ca þ cb aþb 2 rffiffiffiffiffi 1 þ c 4cc2 6c2 c2 ðx þ ctÞ ; c2 > 0; c4 < 0; ¼ cot2 ca þ cb aþb 2 pffiffiffiffiffiffiffiffiffiffiffi 1 þ c þ 8cc2 24c2 2 2c2 ðx þ ctÞ ; c2 < 0; c4 > 0; ¼ þ csc h ca þ cb aþb pffiffiffiffiffiffiffiffi 1 þ c þ 8cc2 24c2 ¼ 2c2 ðx þ ctÞ ; c2 > 0; c4 < 0: csc2 ca þ cb aþb
If c0 ; c2 ; c4 satisfy c2 > 0; c4 < 0; c0 ¼ tions when a þ b – 0:
c22 m2 ð1m2 Þ c4 ð2m2 1Þ2
, from (A.6) and (3.4) we find three Jacobi elliptic doubly periodic type solu-
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rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðx þ ctÞ ; 2m2 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c2 ð1 m2 Þ c2 2 ðx þ ctÞ ; ¼ þ nc ða þ bÞð2m2 1Þ ca þ cb 2m2 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c2 m2 c2 2 ðx þ ctÞ ; ¼ þ cn ca þ cb ða þ bÞð2m2 1Þ 2m2 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 12c2 ð1 m2 Þ c2 þ nc2 ðx þ ctÞ : 2 2 ða þ bÞð2m 1Þ 2m 1
u27 ¼ u28 u29
1 þ c 4cc2 12c2 m2 þ cn2 ca þ cb ða þ bÞð2m2 1Þ
c2 ð1m2 Þ
If c0 ; c2 ; c4 satisfy c2 > 0; c4 < 0; c0 ¼ c 2ð2m2 Þ2 , from (A.7) and (3.4) we find three Jacobi elliptic doubly periodic type solutions 4 when a þ b – 0:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðx þ ctÞ ; 2 m2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c22 ð1 m2 Þ c2 2 ðx þ ctÞ ; ¼ nd þ ða þ bÞð2 m2 Þm2 ca þ cb 2 m2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12m2 c2 2 ðx þ ctÞ ; ¼ þ dn ca þ cb ða þ bÞð2 m2 Þ 2 m2 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12c22 ð1 m2 Þ c2 2 þ nd ðx þ ctÞ : 2 2 2 ða þ bÞð2 m Þm 2m
u210 ¼ u211 u212
1 þ c 4cc2 12m2 2 þ dn ca þ cb ða þ bÞð2 m2 Þ
c2 m2
If c0 ; c2 ; c4 satisfy c2 < 0; c4 > 0; c0 ¼ c ðm22 þ1Þ2 , from (A.8) and (3.4) we find three Jacobi elliptic doubly periodic type solutions 4 when a þ b – 0:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðx þ ctÞ ; 2 m þ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c2 c2 2 ðx þ ctÞ ; ¼ þ ns ca þ cb ða þ bÞðm2 þ 1Þ m2 þ 1 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ c 4cc2 12c2 m2 c2 ðx þ ctÞ ; 2 ¼ þ sn2 2 ca þ cb ða þ bÞðm þ 1Þ m þ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12c2 c2 2 ðx þ ctÞ : þ ns ða þ bÞðm2 þ 1Þ m2 þ 1
u213 ¼ u214 u215
1 þ c 4cc2 12c2 m2 þ sn2 ca þ cb ða þ bÞðm2 þ 1Þ
Case 3. c0 ¼ c1 ¼ c4 ¼ 0
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼
b1 c 2 ; c3
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 0;
1 ; 4 c2 1 1 c¼ : c2 1 c¼
From (A.9)–(A.11) and (3.4), we find two hyperbolic type solutions, two triangular type solutions and two polynomial type solutions when a þ b ¼ 0:
b1 c 3 1 2 pffiffiffiffiffi sinh c2 x þ t c2 > 0; 4c2 1 4c2 b1 c 3 1 2 pffiffiffiffiffiffiffiffiffi ¼ a0 sin c2 x þ t c2 < 0; 4c2 1 4c2
u31 ¼ a0 þ u32
u33 ¼ a0 þ b1 c3 ðx tÞ2 þ b2 c23 ðx tÞ4 c2 ¼ 0; pffiffiffiffiffi b1 c 3 1 c2 2 cosh xþ t c2 > 0; u34 ¼ a0 c2 1 2 c2 pffiffiffiffiffiffiffiffiffi c2 b1 c 3 1 cos2 xþ u35 ¼ a0 t c2 < 0; 2 c2 1 c2 u36 ¼ a0 þ b1 c3 ðx tÞ2
c2 ¼ 0:
Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
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Case 4. c2 ¼ c4 ¼ 0; c0 – 0; c1 – 0; c3 > 0
a2 ¼ 0;
b1 ¼ 0;
a1 ¼ 0;
a2 ¼ 0;
1þc 3c3 ; a1 ¼ ; ca þ cb aþb 4 c0 þ 3 c c21 þ 4 cc0 6 c1 a0 ¼ ; b1 ¼ ; 4 a cc0 þ 4 b cc0 aþb b2 ¼ 0;
a0 ¼
b2 ¼
12 c0
aþb
;
c3 ¼
c31 : 8 c20
From (A.12) and (3.4) we find two Weierstrass elliptic doubly periodic type solutions when a þ b – 0:
pffiffiffiffiffi 1þc 3c3 c3 ðx þ ctÞ; g 2 ; g 3 ; } ca þ cb a þ b 2 3cc21 þ 4cc0 þ 4c0 6c pffiffiffiffi 1 ¼ c3 4cc0 a þ 4cc0 b ða þ bÞ} 2 ðx þ ctÞ; g 2 ; g 3
u41 ¼ u42
12c0 pffiffiffiffi 2 ; c3 ða þ bÞ } 2 ðx þ ctÞ; g 2 ; g 3
where g 2 ¼ 4c1 =c3 and g 3 ¼ 4c0 =c3 . Case 5. c3 ¼ c4 ¼ 0; c0 ¼ c21 =4c2 ; c2 > 0
a ¼ b; a2 ¼ 0; b2 ¼ 0; c ¼
1 ; c2 1
c1 ¼ 0;
a1 c2 1 ; c¼ ; 4 c2 1 c1 6c1 12c0 b1 ¼ ; b2 ¼ ; aþb aþb
a ¼ b; b1 ¼ 0; b2 ¼ 0; b1 ¼ 0; a2 ¼ a1 ¼ 0;
a2 ¼ 0;
a0 ¼
1 þ c cc2 ; ca þ cb
c ¼ 1:
From (A.17) and (3.4), we find two exponential type solutions when a þ b ¼ 0 and an exponential type solution when a þ b – 0:
pffiffiffiffiffi pffiffiffiffiffi e c2 u51 ¼ a0 þ b1 exp e c2 x t ; c2 1 pffiffiffiffiffi pffiffiffiffiffi 2 pffiffiffiffiffi pffiffiffiffiffi a1 c1 e c2 a1 c2 c1 e c2 t þ t þ a1 exp e c2 x þ exp e c2 x þ ; u52 ¼ a0 4c2 1 4c2 1 2c2 c1 2c2 when a þ b ¼ 0.
u53 ¼
pffiffiffiffiffi pffiffiffiffiffi 1 12c0 pffiffiffiffiffi pffiffiffiffiffi 2 c2 6c1 c1 c1 þ exp ðe c2 x e c2 tÞ þ exp ðe c2 x e c2 t Þ ; þ aþb aþb 2c2 aþb 2c2
when a þ b – 0. Case 6. c0 ¼ c3 ¼ c4 ¼ 0
a ¼ b; b1 ¼ 0; b2 ¼ 0; a2 ¼
a1 c2 ; c1
c¼
1 ; 4 c2 1
1 ; c2 1 1 þ c c c2 3c1 ; b1 ¼ : a0 ¼ aþb aþb
a ¼ b; a2 ¼ 0; b1 ¼ 0; b2 ¼ 0; c ¼ a1 ¼ 0;
a2 ¼ 0;
b2 ¼ 0;
From (A.18), (A.19) and (3.4), we find two triangular type solutions and two hyperbolic type solutions when a þ b ¼ 0, a triangular type solution and a solitary wave solution when a þ b – 0:
2 pffiffiffiffiffiffiffiffiffi a1 c1 a1 c1 1 2 e sin c2 x þ c2 < 0; t c2 1 4c2 4c2 2 pffiffiffiffiffi a1 c1 a1 c1 1 ¼ a0 þ 2 e sinh 2 c2 x þ c2 > 0; t c2 1 4c2 4c2 pffiffiffiffiffiffiffiffiffi a1 c1 1 ¼ a0 1 e sin c2 x þ t c2 < 0; c2 1 2c2 pffiffiffiffiffi a1 c1 1 t c2 > 0; ¼ a0 1 e sinh 2 c2 x þ c2 1 2c2
u61 ¼ a0 þ u62 u63 u64
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Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
when a þ b ¼ 0.
pffiffiffiffiffiffiffiffiffi 1 þ c þ cc2 6c2 1 ð1 e sin ð c2 ðx þ ctÞÞÞ c2 < 0; þ ca þ cb aþb pffiffiffiffiffi 1 þ c þ cc2 6c2 1 ¼ c2 > 0; ð1 e sinh ð2 c2 ðx þ ctÞÞÞ þ ca þ cb aþb
u65 ¼ u66
when a þ b – 0. Case 7. c0 ¼ c1 ¼ 0; c2 – 0; c3 – 0; c4 > 0
1 ; c2 1 b c 1 a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 1 2 ; c ¼ ; 4 c2 1 c3 1 þ c cc2 6c3 12c4 b1 ¼ 0; b2 ¼ 0; a0 ¼ ; a1 ¼ ; a2 ¼ ; ca þ cb aþb aþb
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 0; c ¼
c23 ¼ 4c1 c2 :
From (A.22), (A.23), (A.24) and (3.4), we find three hyperbolic type solutions and three triangular type solutions when a þ b ¼ 0, and a kink shaped solitary wave solution when a þ b – 0:
u71
u72 u73
u74
u75 u76
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2eb1 c2 c4 tan 12 c2 x þ c211 t þ b1 c3 pffiffiffiffiffiffiffiffiffi ¼ a0 c2 < 0; c2 sec2 12 c2 ðx þ c211 tÞ pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2eb1 c2 c4 tanh 12 c2 x þ c211 t b1 c3 pffiffiffiffiffi c2 > 0; ¼ a0 þ 2 c2 sech 12 c2 x þ c211 t pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2b1 c4 pffiffiffiffiffi c2 > 0; c3 ¼ 2e c2 c4 ; ¼ a0 þ pffiffiffiffiffi 1 1 e c2 1 þ tanh 2 c2 x þ c2 1 t pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2eb1 c2 c4 tan 12 c2 x þ 4c211 t þ b1 c3 b1 2e c2 c4 tan 12 c2 x þ 4c211 t þ c3 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi þ c2 < 0; ¼ a0 c2 sec2 12 c2 x þ 4c211 t c3 c2 sec4 12 c2 x þ 4c211 t pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2eb1 c2 c4 tanh 12 c2 x þ 4c211 t b1 c3 b1 ð2e c2 c4 tanh 12 c2 x þ 4c211 t c3 Þ2 pffiffiffiffiffi pffiffiffiffiffi ¼ a0 þ þ c2 > 0; 2 4 c2 sech 12 c2 x þ 4c211 t c3 c2 sech 12 c2 x þ 4c211 t pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2b1 c4 4b1 c4 pffiffiffiffiffi þ ¼ a0 þ pffiffiffiffiffi pffiffiffiffiffi 2 c2 > 0; c3 ¼ 2e c2 c4 ; 1 1 e c2 1 þ tanh 12 c2 x þ 4c211 t c3 1 þ tanh 2 c2 x þ 4c2 1 t
when a þ b ¼ 0; and
u77 ¼
2 1 þ c þ cc2 6c2 1 pffiffiffiffiffi 3c2 1 pffiffiffiffiffi 1 þ tanh 1 þ tanh c2 ðx þ ctÞ c2 ðx þ ctÞ 2 2 ca þ cb aþb aþb
c2 > 0;
pffiffiffiffiffiffiffiffiffi c3 ¼ 2e c2 c4
when a þ b – 0. Case 8. c0 ¼ c1 ¼ c2 ¼ 0; c4 < 0
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 0; c ¼ 1: From (A.16) and (3.4), we find an exponential type solution when a þ b ¼ 0 as follow:
u81 ¼ a0 þ
2b1 c4 ec3 exp pffiffiffiffiffiffiffiffiffi ðx tÞ : c3 2 c4
Remark 1. As m ! 1 and m ! 0, the Jacobi elliptic doubly periodic type solutions u27 u215 can be reduced to u21 u23 through proper transformation. The solutions u21 u26 and u17 have been found by Elwakil [12] (by proper transformation). The other solutions we have found are all new. We also find the solutions when a þ b ¼ 0, which was not given by Elwakil. Remark 2. For illustration, the solutions for the GSWW equation are displayed in Figs. 1–3 for the cases when a þ b ¼ 0 and Figs. 4–7 for the cases when a þ b – 0.
Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
Fig. 1. Hyperbolic type solution u11 with a0 ¼ 1; b1 ¼ 1; c2 ¼ 2; c4 ¼ 1, and u64 with a0 ¼ 1; a1 ¼ 1; c1 ¼ 1; c2 ¼ 2.
Fig. 2. Exponential type solution u52 with a0 ¼ 2; a1 ¼ 1; c1 ¼ 1; c2 ¼ 1, and u76 with a0 ¼ 1; b1 ¼ 1; c2 ¼ 2; c4 ¼ 1.
Fig. 3. Triangular type solution u12 with a0 ¼ 1; b1 ¼ 1; c2 ¼ 1; c4 ¼ 1 and polynomial type solution u14 with a0 ¼ 1; b1 ¼ 1; b2 ¼ 1; c4 ¼ 1.
Fig. 4. Solitary wave solution u21 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1, and u23 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 2.
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Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
Fig. 5. Solitary wave solution u53 with a ¼ 0:5; b ¼ 0:5; c0 ¼ 1; c1 ¼ 1; c2 ¼ 1; and u66 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1.
Fig. 6. Solitary wave solution u77 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1; and triangular type solution u22 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1.
Fig. 7. Jocobi elliptic doubly periodic type solution u213 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1=9; m ¼ 0:8; and Weierstass elliptic doubly periodic type solution u42 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c0 ¼ 0:1; c1 ¼ 0:2; c2 ¼ 0:1.
4. Conclusion In this paper we use a rational expansion to improve the Fan’s method for solving the GSWW equation. We successfully obtain kinds of new exact solutions of the GSWW equation by the proposed method. The new solutions, for examples, kink shaped solitary solution u77 , Jacobi elliptic doubly periodic type solution u27 , Weierstrass elliptic doubly periodic type solution u41 , exponential type solution u53 and triangular type solution u65 have never been found before. More nonlinear PDEs can be expected to be solved by this method. Acknowledgements The work described in this paper was fully supported by a Grant from CityU (Project No. 7001791).
Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
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Appendix A The fundamental solutions of x0 ¼ e follows:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c0 þ c1 x þ c2 x2 þ c3 x3 þ c4 x4 for different choices of c0 ; c1 ; c2 ; c3 are listed as
Case 1. c0 ¼ c1 ¼ c3 ¼ 0 A bell shaped solitary wave solution, a triangular type solution and a rational solution
rffiffiffiffiffiffiffiffiffi pffiffiffiffiffi c x ¼ 2 sechð c2 nÞ; c2 > 0; c4 < 0; c4 rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi c x ¼ 2 sec ð c2 nÞ; c2 < 0; c4 > 0; c4
e x ¼ pffiffiffiffiffi ; c2 ¼ 0; c4 > 0: c4 n
ðA:1Þ ðA:2Þ ðA:3Þ
Case 2. c1 ¼ c3 ¼ 0 A kink shaped solitary wave solution, a triangular type solution and three Jacobi elliptic doubly periodic type solutions
rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi c2 c2 c2 tanh n ; c2 < 0; c4 > 0; c0 ¼ 2 ; 2c4 2 4c4 rffiffiffiffiffiffiffiffi rffiffiffiffiffi c c2 c2 n ; c2 > 0; c4 > 0; c0 ¼ 2 ; x ¼ e 2 tan 2c4 2 4c4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c2 m c2 c2 m2 ð1 m2 Þ n ; c2 > 0; c4 < 0; c0 ¼ 2 x¼ ; cn c4 ð2m2 1Þ 2m2 1 c4 ð2m2 1Þ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 c2 c2 ð1 m2 Þ dn x¼ ; n ; c2 > 0; c4 < 0; c0 ¼ 2 2 2 c4 ð2 m Þ 2m c4 ð2 m2 Þ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 m2 c2 c22 m2 n ; c2 < 0; c4 > 0; c0 ¼ x¼e ; sn 2 2 c4 ðm þ 1Þ m þ1 c4 ðm2 þ 1Þ2
x¼e
ðA:4Þ ðA:5Þ ðA:6Þ ðA:7Þ ðA:8Þ
where m is a modulus. Case 3. c0 ¼ c1 ¼ c4 ¼ 0 A bell shaped wave solitary solution, a triangular type solution and a rational type solution
pffiffiffiffiffi c2 c2 2 sech n ; c2 > 0; 2 c3 pffiffiffiffiffiffiffiffiffi c c2 x ¼ 2 sec2 n ; c2 < 0; c3 2 1 x ¼ 2 ; c2 ¼ 0: c3 n
x¼
ðA:9Þ ðA:10Þ ðA:11Þ
Case 4. c2 ¼ c4 ¼ 0; c0 – 0; c1 – 0; c3 > 0 A Weierstrass elliptic doubly periodic type solution
pffiffiffiffiffi c3 n; g 2 ; g 3 ; 2
x¼}
ðA:12Þ
where g 2 ¼ 4c1 =c3 and g 3 ¼ 4c0 =c3 are called invariants of Weierstrass elliptic function. Case 5. c2 ¼ c3 ¼ c4 ¼ 0 Two polynomial type solutions
pffiffiffiffiffi
x ¼ e c0 n; c1 ¼ 0; c0 > 0;
ðA:13Þ
c 1 x ¼ 0 þ c 1 n2 ; c1 4
ðA:14Þ
c1 – 0:
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Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
Case 6. c0 ¼ c1 ¼ c2 ¼ 0 A rational type solution and a exponential type solution
4c3 ; c4 – 0; c23 n2 4c4 c ec3 ffin ; x ¼ 3 exp pffiffiffiffiffiffiffiffi 2c4 2 c4
x¼
ðA:15Þ c4 < 0:
ðA:16Þ
Case 7. c3 ¼ c4 ¼ 0 A exponential type solution, a triangular type solution and a hyperbolic type solution
pffiffiffiffiffi c1 c2 þ exp ðe c2 nÞ; c2 > 0; c0 ¼ 1 ; 2c2 4c2 pffiffiffiffiffiffiffiffiffi c ec x ¼ 1 þ 1 sin ð c2 nÞ; c0 ¼ 0; c2 < 0; 2c2 2c2 pffiffiffiffiffi c ec x ¼ 1 þ 1 sinh ð2 c2 nÞ; c0 ¼ 0; c2 > 0 2c2 2c2 rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi c x ¼ e 0 sin ð c2 nÞ; c1 ¼ 0; c0 > 0; c2 < 0; c2 rffiffiffiffiffi pffiffiffiffiffi c x ¼ e 0 sinh ð c2 nÞ; c1 ¼ 0; c0 > 0; c2 > 0: c2
x¼
ðA:17Þ ðA:18Þ ðA:19Þ ðA:20Þ ðA:21Þ
Case 8. c0 ¼ c1 ¼ 0; c4 > 0 A triangular type solution, two solitary wave solutions
pffiffiffiffiffiffiffiffiffi c2 sec2 12 c2 n pffiffiffiffiffiffiffiffiffi x ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; c2 < 0; 2e c2 c4 tan 12 c2 n þ c3 p ffiffiffiffi ffi 2 pffiffiffiffiffiffiffiffiffi c2 sech ð12 c2 nÞ pffiffiffiffiffi x ¼ pffiffiffiffiffiffiffiffiffi ; c2 > 0; c3 – 2e c2 c4 ; 2e c2 c4 tanh 12 c2 n c3 rffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 1 c 1 pffiffiffiffiffi x ¼ e 2 1 þ tanh c2 n ; c2 > 0; c3 ¼ 2e c2 c4 : 2 2 c4
ðA:22Þ ðA:23Þ ðA:24Þ
The Jacobi elliptic functions are doubly periodical and possess properties of triangular functions: 2
sn2 n þ cn2 n ¼ 1;
dn n ¼ 1 m2 sn2 n;
ðsnnÞ0 ¼ cnndnn;
ðcnnÞ0 ¼ snndnn;
ðdnnÞ0 ¼ m2 snncnn:
When m ! 1, the Jacobi functions degenerate to the following hyperbolic functions:
snn ! tanh n;
cnn ! sech n;
dnn ! sech n
and when m ! 0, the Jacobi functions degenerate to the following triangular functions:
snn ! sin n;
cnn ! cos n;
dnn ! 1:
A more detail notations for the Weierstrass and Jacobi elliptic functions can be found in [35,36]. When m ! 1, the Jacobi doubly periodic solutions (A.6) and (A.7) degenerate to the solitary wave solutions (A.1) and the solution (A.8) degenerates to (A.4). References [1] A. Espinosa, J. Fujioka, Hydrodynamic foundation and Pailevé analysis of the Hirota–Satsuma-type equations, J. Phys. Soc. Jpn. 63 (1994) 1289–1294. [2] P.A. Clarkson, E.L. Mansfield, On a shallow water wave equation, Nonlinearity 7 (1994) 975–1000. [3] M.J. Ablowitz, D.J. Kaup, A.C. Newell, Segur, The inverse scattering transform: fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974) 249– 315. [4] R. Hirota, J. Satsuma, N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Jpn. 40 (1976) 611–612. [5] P.A. Clarkson, E.L. Mansfield, Symmetry reductions and exact solutions of shallow water wave equations, Acta Appl. Math. 39 (1995) 245–276. [6] J. Weiss, M. Tabor, G. Carnevale, The Pailevé property for partial differential equations, J. Math. Phys. 24 (1983) 522–526. [7] M.J. Ablowitz, A. Ramani, H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type, I, J. Math. Phys. 21 (1980) 715–721.
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[8] J. Hietarinta, Hirota’sbilinear methhod and partial integrability, in: R. Conte, N. Bocarra (Eds.), Partially Integrable Evolution Equations in Physics, NATO ASI Series C: Mathematical and Physical Sciences, vol. 310, Kluwer, Dordrecht, 1990, pp. 459–478. [9] R. Hirota, Direct methods in soliton theory, in: R.K. Bullough, Caudrey (Eds.), Solitons, Topics in Current Physics, vol. 17, Springer-Verlag, Berlin, 1980, pp. 157–176. [10] G.W. Bluman, J.D. Cole, The general similarity of the heat equation, J. Math. Mech. 18 (1969) 1025–1042. [11] P.A. Clarkson, M.D. Kruskal, New similirity solutions of the Boussinesq equation, J. Math. Phys. 30 (1989) 2201–2213. [12] S.A. Elwalil, S.K. El-Labany, M.A. Zahran, R. Sabry, Modified extended tanh-function method and its applications to nonlinear equation, Appl. Math. Comput. 161 (2005) 403–412. [13] M.J. Ablowitz, P.A. Clarkson, Solitons, nonlinear evolution evolution equations and inverse scattering, Cambridge University Press, Cambridge, 1991. [14] R. Beals, R.R. Coifman, Scattering and inverse scattering for 1st order system, Commun. Pure. Appl. Math. 37 (1984) 39–90. [15] V.B. Matveev, M.A. Salle, Darboux Transformation and Solitons, Springer, Berlin, 1991. [16] C.H. Gu, H.S. Hu, Z.X. Zhou, Darboux Transformations in Soliton Theory and its Geometric Applications, Shanghai Science Technology Publisher, Shanghai, 1999. [17] S.B. Leble, N.V. Ustinov, Darboux transformations, deep reductions and solitons, J. Phys. A. 26 (1993) 5007–5016. [18] P.G. Esteevez, Darboux transformation and solutions for an equation in 2 + 1 dimensions, J. Math. Phys. 40 (1999) 1406–1419. [19] V.G. Dubrousky, B.G. Konopelchenko, Delta-dressing and exact solutions for the (2 + 1)-dimensional Harry Dym equation, J. Phys. A. 27 (1994) 4719– 4721. [20] E.G. Fan, A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations, J. Math. Phys. 42 (2001) 4327– 4344. [21] R. Hirta, J. Satsuma, Soliton solution of a coupled KdV equation, Phys Lett. A. 85 (1981) 407–408. [22] H.W. Tam, W.X. Ma, X.B. Hu, D.L. Wang, The Hirota–Satsuma coupled KdV equation and a coupled Ito system revisited, J. Phys. Soc. Jpn. 69 (2000) 45– 51. [23] F. Cariello, M. Tabor, Pailevé expansions for nonintegtable evolution equations, Physica. D. 39 (1989) 77–94. [24] A. Biswas, 1-Soliton solution of the B(m, n) equation with generalized evolution, Commun. Nonlinear Sci. Numer. Simulation 14 (2009) 3226–3229. [25] A. Biswas, D. Milovic, A. Ranasinghe, Solitary waves of Boussinesq equation in a power law media, Commun. Nonlinear Sci. Numer. Simulation 14 (2009) 3738–3742. [26] M.L. Wang, Solitary wave solutions for variant Boussinesq equation, Phys. Lett. A. 199 (1995) 169–172. [27] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992) 650–654. [28] W. Hereman, Exact solutions to the two-dimensional Korteweg–de Vries–Burgers equations using MACSYMA, Comput. Phys Commun. 65 (1996) 143– 150. [29] E.J. Parkes, Exact solutions to the two-dimensional Korteweg–de Vries–Burgers equation, J. Phys. A. 27 (1994) 497–502. [30] E.J. Parkes, B.R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun. 98 (1996) 288–300. [31] E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A. 277 (2000) 212–218. [32] E.G. Fan, Travelling wave solutions for nonlinear equations using symbolic computation, Comput. Math. Appl. 43 (2002) 671–680. [33] S.A. Elwakil, S.K. El-labany, M.A. Zahran, R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A. 299 (2002) 179–188. [34] X.D. Zheng, Y. Chen, H.Q. Zhang, Generalized extended tanh-function method and its application to (1 + 1)-dimensional dispersive long wave equation, Phys. Lett. A. 311 (2003) 145–157. [35] N.L. Akhiezer, Elements of theory of elliptic functions, Am. Math. Soc. Providence (1990). [36] Z.X. Wang, X.J. Xia, Special functions, World Scientific, 1989.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A rational generalization of Fan’s method and its application to generalized shallow water wave equations Zonghang Yang a,*, Y.C. Hon b a b
Department of Information Systems, City University of Hong Kong, Hong Kong SAR, PR China Department of Mathematics, City University of Hong Kong, Hong Kong SAR, PR China
a r t i c l e
i n f o
a b s t r a c t In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations. Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved.
Keywords: The GSWW equation Fan’s method Rational expansion Travelling wave solution
1. Introduction The Boussinesq approximation theory [1] for generalized classical shallow water wave leads to the GSWW equation:
v xxxt þ av x v xt þ bv t v xx v xt v xx ¼ 0; where a and b are non-zero constants. Using
uxxt þ auut
bux @ 1 x ut
ð1:1Þ
v x ¼ u in [2], Eq. (1.1) can be simplified to
ut ux ¼ 0;
ð1:2Þ
which has recently attracted many attentions from researchers under the following cases: (1) For the case when a ¼ 2b, the following AKNS–SWW equation
uxxt þ 2buut bux @ 1 x ut ut ux ¼ 0;
ð1:3Þ
was discussed by Ablowitz et al. [3]. (2) For the case when a ¼ b, the following equation
uxxt þ buut bux @ 1 x ut ut ux ¼ 0;
ð1:4Þ
was discussed by Hirota and Satsuma [4]. The GSWW Equation (1.1) in potential form was studied by Clarkson and Mansfield [2,5] who gave a complete catalog of classical and nonclassical symmetry reductions. The necessary conditions of Painlevé tests was given by Ablowitz et al. [6] and the complete integrability of Eq. (1.1) for the case when a ¼ 2b or a ¼ b was established by Weiss et al. [7]. It had been proven by Hietarinta [8] that the GSWW Equation (1.2) can be expressed in Hirota’s bilinear form and when a ¼ 2b, Eq. (1.2)
* Corresponding author. E-mail addresses: [email protected], [email protected] (Z. Yang), [email protected] (Y.C. Hon). 0096-3003/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.03.029
Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
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can be reduced to (1.3) or to (1.4) when a ¼ b. In other words, both Eqs. (1.3) and (1.4) are solvable by using Hirota’s bilinear method [9]. The N-soliton solutions for Eqs. (1.3) and (1.4) had also been found using this technique [4]. Although there is no scaling transformation that can reduce (1.3) to (1.4), the classical methods of Lie, the nonclassical method of Bluman and Cole, and the direct method of Clarkson and Kruskal can be applied to solve the GSWW Equation (1.2) to obtain some kinds of symmetry reductions [10,11]. Recently, Elwakil found a lot of exact solutions by using the modified extended tanh-function method [12]. Since nonlinear PDEs are widely used to describe complex phenomena in various field of sciences, various methods for obtaining exact travelling solitary wave solutions to nonlinear evolution equations have been proposed. For instances, inverse scattering method [13,14], Darboux transformation [15–20], Hirota bilinear method [21,22], Painleve’ expansions [23], solitary wave ansatze method for Boussinesq equation [24,25], Homogeneous balance method [26] and tanh-function method [27–34]. Among these methods, the tanh-function method provides a straightforward and effective algorithm to obtain particular solutions for system of nonlinear partial differential equations by taking the advantage that solitary solutions are essentially of localized nature. Based on the solution to the well-known Riccati equation, Fan presented a useful extended tanh-function method [31] and further extended this method to solve many different kinds of partial differential equations [32]. Due to the importance of Fan’s generalization, this method is also named as Fan’s method. Recently, Elwakil presented a modified extended tanh-function method [33] and Zheng, Chen and Zhang developed a generalized extended tanh-function method [34]. In this paper we devised a rational generalization for Fan’s method and successfully solved GSWW Eq. (1.2). To the acknowledge of the authors, some of the obtained solutions have never been given before. 2. Methodology To illustrate the idea of our proposed method, we first consider the following ordinary differential equation (ODE)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x0 ¼ e c0 þ c1 x þ c2 x2 þ c3 x3 þ c4 x4 ;
ð2:1Þ
where x ¼ xðnÞ, and e ¼ 1, 0 denotes d=dn. For different choices of c0 ; c1 ; c2 ; c3 and c4 , Eq. (2.1) admits different kinds of fundamental solutions which can be used to construct the exact solution of nonlinear PDEs. The details of the solution of Eq. (2.1) are presented in Appendix A. Consider the following nonlinear partial differential equation
Hðu; ut ; ux ; uxx ; . . .Þ ¼ 0:
ð2:2Þ
The main steps of our proposed method are outlined as follows. Step 1. By using the wave transformation uðx; tÞ ¼ UðnÞ; n ¼ x þ ct (c is a non-zero constant), we reduce Eq. (2.2) into the following ODE
HðU; U 0 ; U 00 ; . . .Þ ¼ 0;
ð2:3Þ
0
where U ¼ dU=dn. Step 2. Introduce a new variable x ¼ xðnÞ which is a solution of the following ODE
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u r x ¼ et cj xj ; 0
ð2:4Þ
j¼0
where cj ; ðj ¼ 0; . . . ; rÞ are constants to be determined. The high order derivatives with respect to the variable n become:
x00 ¼
r eX
2
jcj xj1 ; . . .
ð2:5Þ
j¼1
Step 3. By virtue of the new variable x, we expand the solution of Eq. (2.3) as
uðx; tÞ ¼ UðnÞ ¼
n X
ai xi þ
i¼0
n X
bi xi ;
ð2:6Þ
i¼1
where ai ði ¼ 0; 1; . . . ; nÞ; bi ði ¼ 1; 2; . . . ; nÞ are constants to be determined. This expansion develops the polynomial expansion in Fan’s method [31,32] as
uðx; tÞ ¼ UðnÞ ¼
n X i¼0
ai xi ;
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Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
by adding rational terms. To determine the values of n and r, we substitute (2.4) and (2.5) into (2.3) and balance the highest order linear term with the nonlinear terms in (2.3) to obtain a relation for n and r. For illustration, in solving the following Burgers’ equation
ut þ uux uxx ¼ 0; balancing the terms uux and uxx we obtain
1 n þ n 1 þ r ¼ n 2 þ r; 2
ð2:7Þ
and hence 2n ¼ r 2. The values of n and r then lead to the series expansion of the travelling wave solutions for (2.2). For instance, if we take n ¼ 1 and r ¼ 4 in (2.7), we can use the following expansion as a solution of the Burgers’ equation
u ¼ a0 þ a1 x þ b1 x1 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x0 ¼ e c0 þ c1 x þ c2 x2 þ c3 x3 þ c4 x4 : If we take n ¼ 2 and r ¼ 6 in (2.7), we have
u ¼ a0 þ a1 x þ a2 x þ b1 x1 þ b2 x2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x0 ¼ e c0 þ c1 x þ c2 x2 þ c3 x3 þ c4 x4 þ c5 x5 þ c6 x6 : We can then take r ¼ 4 and make use of the fundamental solutions of the ODE to derive the solutions of (2.2). Step 4. Substituting the expansions (2.4) and (2.5) into Eq. (2.3) and setting the coefficients of all powers of xi and qP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r i j x j¼0 cj x to zero, we obtain a system of algebraic equations, from which the constants ai ; ði ¼ 0; 1; . . . ; nÞ; bi ; ði ¼ 1; 2; . . . ; nÞ; cj ; ðj ¼ 0; 1; . . . ; rÞ can be found explicitly by using symbolic software such as mathematica. Step 5. Analyzing the parameters cj ; ðj ¼ 0; 1; . . . ; rÞ obtained in Step 4, we can then construct many kinds of solutions if (2.3) is of exact solvability. In this paper we only consider the case when r ¼ 4 due to the efficiency in using the fundamental solutions. It is noted here that the solution of the system of algebraic equations will become tedious with an increase of the values of n and r. Substituting ai ; ði ¼ 0; 1; . . . ; nÞ; bi ; ði ¼ 1; 2; . . . ; nÞ and the fundamental solutions of Eq. (2.3) into (2.6), we can now obtain the exact travelling solutions for Eq. (2.2). 3. Exact solutions for the GSWW equation Using the wave transformations u ¼ UðnÞ; n ¼ x þ ct, we transform Eq. (1.2) to the following ODE
cU 000 þ cða þ bÞUU 0 ðc þ 1ÞU ¼ 0:
ð3:1Þ
From (2.6) we expand the solution of (3.1) as
U¼
n X
a i xi þ
i¼0
n X
bi xi ;
ð3:2Þ
i¼1
where x satisfies (2.4). Balancing the highest order linear terms with the nonlinear terms in (3.1) gives
3 1 n 3 þ r ¼ n þ n 1 þ r: 2 2
ð3:3Þ
From (3.3) we get the relation n ¼ r 2 which we choose n ¼ 2 and r ¼ 4 to obtain
U ¼ a0 þ a1 x þ a2 x2 þ b1 x1 þ b2 x2 ;
ð3:4Þ i
where x satisfies (2.1). Substituting (3.4), (2.4) and (2.5) into (3.1) and setting the coefficients of all powers like x and qP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r j xi j¼0 cj x to zero, we then obtain a system of algebraic equations with the aid of Mathematica. The system of algebraic equations is given as follows 2
2
2 c a e b2 þ 2 c b e b2 þ 24 c e3 b2 c0 ¼ 0; 3 c a e b1 b2 þ 3 c b e b1 b2 þ 6 c e3 b1 c0 þ 15 c e3 b2 c1 ¼ 0; 2
2
c a e b1 þ c b e b1 2 e b2 2 c e b2 þ 2 c a e a0 b2 þ 2 c b e a0 b2 þ 3 c e3 b1 c1 þ 8 c e3 b2 c2 ¼ 0; e b1 c e b1 þ c a e a0 b1 þ c b e a0 b1 þ c a e a1 b2 þ c b e a1 b2 þ c e3 b1 c2 þ 3 c e3 b2 c3 ¼ 0;
e a1 þ c e a1 c a e a0 a1 c b e a0 a1 c a e a2 b1 c b e a2 b1 3 c e3 a2 c1 c e3 a1 c2 ¼ 0; c a e a21 c b e a21 þ 2 e a2 þ 2 c e a2 2 c a e a0 a2 2 c b e a0 a2 8 c e3 a2 c2 3 c e3 a1 c3 ¼ 0; 3 c a e a1 a2 3 c b e a1 a2 15 c e3 a2 c3 6 c e3 a1 c4 ¼ 0; 2 c a e a22 2 c b e a22 24 c e3 a2 c4 ¼ 0:
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Solving the above algebraic system with the aid of Mathematica, we list the solutions of the system under the following several cases: Case 1. c0 ¼ c1 ¼ c3 ¼ 0; c4 – 0
1 ; c2 1 a ¼ b; a1 ¼ 0; a2 ¼ 0; c ¼ 1; c2 ¼ 0; 1 þ c 4 c c2 12 c4 a1 ¼ 0; b1 ¼ 0; b2 ¼ 0; a0 ¼ ; a2 ¼ : ca þ cb aþb
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 0; c ¼
From (A.1), (A.2), (A.3) and (3.4), we find a hyperbolic type solution, a triangular type solution and two polynomial type solutions when a þ b ¼ 0, a bell shaped solitary wave solution, a triangular type solution and a rational type solution when a þ b – 0:
rffiffiffiffiffiffiffiffiffi pffiffiffiffiffi c4 1 cosh c2 x þ t c2 > 0; c4 < 0; c2 1 c2 rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi c4 1 c2 x þ ¼ a0 þ b1 cos t c2 < 0; c4 > 0; c2 1 c2 pffiffiffiffiffi b1 c 4 ¼ a0 ðx tÞ c2 ¼ 0; c4 > 0; e ffiffiffiffiffi p b1 c4 ðx tÞ ¼ a0 þ þ b2 c4 ðx tÞ2 c2 ¼ 0; c4 > 0;
u11 ¼ a0 þ b1 u12 u13 u14
e
when a þ b ¼ 0, and
1 þ c 4cc2 12c2 2 pffiffiffiffiffi þ sech ð c2 ðx þ ctÞÞ c2 > 0; c4 < 0; ca þ cb aþb pffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c2 ¼ þ sec2 ð c2 ðx þ ctÞÞ c2 < 0; c4 > 0; ca þ cb aþb 1þc 12 ¼ c2 ¼ 0; c4 > 0; ca þ cb ða þ bÞðx þ ctÞ2
u15 ¼ u16 u17
when a þ b – 0. Case 2. c1 ¼ c3 ¼ 0
a1 ¼ 0;
b1 ¼ 0;
a1 ¼ 0;
a2 ¼ 0;
a1 ¼ 0;
b1 ¼ 0;
1 þ c 4 c c2 12 c4 ; a2 ¼ ; ca þ cb aþb 1 þ c 4 c c2 12 c0 b1 ¼ 0; a0 ¼ ; b2 ¼ ; ca þ cb aþb 1 þ c 4 cc2 12 c4 12 c0 a0 ¼ ; a2 ¼ ; b2 ¼ : ca þ cb aþb aþb b2 ¼ 0;
a0 ¼
If c0 ; c2 ; c4 satisfy c0 ¼ c22 =4c4 , from (A.4), (A.5) and (3.4), we find three solitary wave solutions and three triangular type solutions when a þ b – 0:
u21 u22 u23 u24 u25 u26
rffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 6c2 c2 2 ðx þ ctÞ ; c2 < 0; c4 > 0; ¼ þ tanh ca þ cb aþb 2 rffiffiffiffiffi 1 þ c 4cc2 6c2 c 2 ðx þ ctÞ ; c2 > 0; c4 < 0; ¼ tan2 ca þ cb aþb 2 rffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 6c2 c2 2 ¼ ðx þ ctÞ ; c2 < 0; c4 > 0; þ coth ca þ cb aþb 2 rffiffiffiffiffi 1 þ c 4cc2 6c2 c2 ðx þ ctÞ ; c2 > 0; c4 < 0; ¼ cot2 ca þ cb aþb 2 pffiffiffiffiffiffiffiffiffiffiffi 1 þ c þ 8cc2 24c2 2 2c2 ðx þ ctÞ ; c2 < 0; c4 > 0; ¼ þ csc h ca þ cb aþb pffiffiffiffiffiffiffiffi 1 þ c þ 8cc2 24c2 ¼ 2c2 ðx þ ctÞ ; c2 > 0; c4 < 0: csc2 ca þ cb aþb
If c0 ; c2 ; c4 satisfy c2 > 0; c4 < 0; c0 ¼ tions when a þ b – 0:
c22 m2 ð1m2 Þ c4 ð2m2 1Þ2
, from (A.6) and (3.4) we find three Jacobi elliptic doubly periodic type solu-
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rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðx þ ctÞ ; 2m2 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c2 ð1 m2 Þ c2 2 ðx þ ctÞ ; ¼ þ nc ða þ bÞð2m2 1Þ ca þ cb 2m2 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c2 m2 c2 2 ðx þ ctÞ ; ¼ þ cn ca þ cb ða þ bÞð2m2 1Þ 2m2 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 12c2 ð1 m2 Þ c2 þ nc2 ðx þ ctÞ : 2 2 ða þ bÞð2m 1Þ 2m 1
u27 ¼ u28 u29
1 þ c 4cc2 12c2 m2 þ cn2 ca þ cb ða þ bÞð2m2 1Þ
c2 ð1m2 Þ
If c0 ; c2 ; c4 satisfy c2 > 0; c4 < 0; c0 ¼ c 2ð2m2 Þ2 , from (A.7) and (3.4) we find three Jacobi elliptic doubly periodic type solutions 4 when a þ b – 0:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðx þ ctÞ ; 2 m2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c22 ð1 m2 Þ c2 2 ðx þ ctÞ ; ¼ nd þ ða þ bÞð2 m2 Þm2 ca þ cb 2 m2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12m2 c2 2 ðx þ ctÞ ; ¼ þ dn ca þ cb ða þ bÞð2 m2 Þ 2 m2 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12c22 ð1 m2 Þ c2 2 þ nd ðx þ ctÞ : 2 2 2 ða þ bÞð2 m Þm 2m
u210 ¼ u211 u212
1 þ c 4cc2 12m2 2 þ dn ca þ cb ða þ bÞð2 m2 Þ
c2 m2
If c0 ; c2 ; c4 satisfy c2 < 0; c4 > 0; c0 ¼ c ðm22 þ1Þ2 , from (A.8) and (3.4) we find three Jacobi elliptic doubly periodic type solutions 4 when a þ b – 0:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðx þ ctÞ ; 2 m þ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c 4cc2 12c2 c2 2 ðx þ ctÞ ; ¼ þ ns ca þ cb ða þ bÞðm2 þ 1Þ m2 þ 1 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ c 4cc2 12c2 m2 c2 ðx þ ctÞ ; 2 ¼ þ sn2 2 ca þ cb ða þ bÞðm þ 1Þ m þ1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12c2 c2 2 ðx þ ctÞ : þ ns ða þ bÞðm2 þ 1Þ m2 þ 1
u213 ¼ u214 u215
1 þ c 4cc2 12c2 m2 þ sn2 ca þ cb ða þ bÞðm2 þ 1Þ
Case 3. c0 ¼ c1 ¼ c4 ¼ 0
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼
b1 c 2 ; c3
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 0;
1 ; 4 c2 1 1 c¼ : c2 1 c¼
From (A.9)–(A.11) and (3.4), we find two hyperbolic type solutions, two triangular type solutions and two polynomial type solutions when a þ b ¼ 0:
b1 c 3 1 2 pffiffiffiffiffi sinh c2 x þ t c2 > 0; 4c2 1 4c2 b1 c 3 1 2 pffiffiffiffiffiffiffiffiffi ¼ a0 sin c2 x þ t c2 < 0; 4c2 1 4c2
u31 ¼ a0 þ u32
u33 ¼ a0 þ b1 c3 ðx tÞ2 þ b2 c23 ðx tÞ4 c2 ¼ 0; pffiffiffiffiffi b1 c 3 1 c2 2 cosh xþ t c2 > 0; u34 ¼ a0 c2 1 2 c2 pffiffiffiffiffiffiffiffiffi c2 b1 c 3 1 cos2 xþ u35 ¼ a0 t c2 < 0; 2 c2 1 c2 u36 ¼ a0 þ b1 c3 ðx tÞ2
c2 ¼ 0:
Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
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Case 4. c2 ¼ c4 ¼ 0; c0 – 0; c1 – 0; c3 > 0
a2 ¼ 0;
b1 ¼ 0;
a1 ¼ 0;
a2 ¼ 0;
1þc 3c3 ; a1 ¼ ; ca þ cb aþb 4 c0 þ 3 c c21 þ 4 cc0 6 c1 a0 ¼ ; b1 ¼ ; 4 a cc0 þ 4 b cc0 aþb b2 ¼ 0;
a0 ¼
b2 ¼
12 c0
aþb
;
c3 ¼
c31 : 8 c20
From (A.12) and (3.4) we find two Weierstrass elliptic doubly periodic type solutions when a þ b – 0:
pffiffiffiffiffi 1þc 3c3 c3 ðx þ ctÞ; g 2 ; g 3 ; } ca þ cb a þ b 2 3cc21 þ 4cc0 þ 4c0 6c pffiffiffiffi 1 ¼ c3 4cc0 a þ 4cc0 b ða þ bÞ} 2 ðx þ ctÞ; g 2 ; g 3
u41 ¼ u42
12c0 pffiffiffiffi 2 ; c3 ða þ bÞ } 2 ðx þ ctÞ; g 2 ; g 3
where g 2 ¼ 4c1 =c3 and g 3 ¼ 4c0 =c3 . Case 5. c3 ¼ c4 ¼ 0; c0 ¼ c21 =4c2 ; c2 > 0
a ¼ b; a2 ¼ 0; b2 ¼ 0; c ¼
1 ; c2 1
c1 ¼ 0;
a1 c2 1 ; c¼ ; 4 c2 1 c1 6c1 12c0 b1 ¼ ; b2 ¼ ; aþb aþb
a ¼ b; b1 ¼ 0; b2 ¼ 0; b1 ¼ 0; a2 ¼ a1 ¼ 0;
a2 ¼ 0;
a0 ¼
1 þ c cc2 ; ca þ cb
c ¼ 1:
From (A.17) and (3.4), we find two exponential type solutions when a þ b ¼ 0 and an exponential type solution when a þ b – 0:
pffiffiffiffiffi pffiffiffiffiffi e c2 u51 ¼ a0 þ b1 exp e c2 x t ; c2 1 pffiffiffiffiffi pffiffiffiffiffi 2 pffiffiffiffiffi pffiffiffiffiffi a1 c1 e c2 a1 c2 c1 e c2 t þ t þ a1 exp e c2 x þ exp e c2 x þ ; u52 ¼ a0 4c2 1 4c2 1 2c2 c1 2c2 when a þ b ¼ 0.
u53 ¼
pffiffiffiffiffi pffiffiffiffiffi 1 12c0 pffiffiffiffiffi pffiffiffiffiffi 2 c2 6c1 c1 c1 þ exp ðe c2 x e c2 tÞ þ exp ðe c2 x e c2 t Þ ; þ aþb aþb 2c2 aþb 2c2
when a þ b – 0. Case 6. c0 ¼ c3 ¼ c4 ¼ 0
a ¼ b; b1 ¼ 0; b2 ¼ 0; a2 ¼
a1 c2 ; c1
c¼
1 ; 4 c2 1
1 ; c2 1 1 þ c c c2 3c1 ; b1 ¼ : a0 ¼ aþb aþb
a ¼ b; a2 ¼ 0; b1 ¼ 0; b2 ¼ 0; c ¼ a1 ¼ 0;
a2 ¼ 0;
b2 ¼ 0;
From (A.18), (A.19) and (3.4), we find two triangular type solutions and two hyperbolic type solutions when a þ b ¼ 0, a triangular type solution and a solitary wave solution when a þ b – 0:
2 pffiffiffiffiffiffiffiffiffi a1 c1 a1 c1 1 2 e sin c2 x þ c2 < 0; t c2 1 4c2 4c2 2 pffiffiffiffiffi a1 c1 a1 c1 1 ¼ a0 þ 2 e sinh 2 c2 x þ c2 > 0; t c2 1 4c2 4c2 pffiffiffiffiffiffiffiffiffi a1 c1 1 ¼ a0 1 e sin c2 x þ t c2 < 0; c2 1 2c2 pffiffiffiffiffi a1 c1 1 t c2 > 0; ¼ a0 1 e sinh 2 c2 x þ c2 1 2c2
u61 ¼ a0 þ u62 u63 u64
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when a þ b ¼ 0.
pffiffiffiffiffiffiffiffiffi 1 þ c þ cc2 6c2 1 ð1 e sin ð c2 ðx þ ctÞÞÞ c2 < 0; þ ca þ cb aþb pffiffiffiffiffi 1 þ c þ cc2 6c2 1 ¼ c2 > 0; ð1 e sinh ð2 c2 ðx þ ctÞÞÞ þ ca þ cb aþb
u65 ¼ u66
when a þ b – 0. Case 7. c0 ¼ c1 ¼ 0; c2 – 0; c3 – 0; c4 > 0
1 ; c2 1 b c 1 a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 1 2 ; c ¼ ; 4 c2 1 c3 1 þ c cc2 6c3 12c4 b1 ¼ 0; b2 ¼ 0; a0 ¼ ; a1 ¼ ; a2 ¼ ; ca þ cb aþb aþb
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 0; c ¼
c23 ¼ 4c1 c2 :
From (A.22), (A.23), (A.24) and (3.4), we find three hyperbolic type solutions and three triangular type solutions when a þ b ¼ 0, and a kink shaped solitary wave solution when a þ b – 0:
u71
u72 u73
u74
u75 u76
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2eb1 c2 c4 tan 12 c2 x þ c211 t þ b1 c3 pffiffiffiffiffiffiffiffiffi ¼ a0 c2 < 0; c2 sec2 12 c2 ðx þ c211 tÞ pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2eb1 c2 c4 tanh 12 c2 x þ c211 t b1 c3 pffiffiffiffiffi c2 > 0; ¼ a0 þ 2 c2 sech 12 c2 x þ c211 t pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2b1 c4 pffiffiffiffiffi c2 > 0; c3 ¼ 2e c2 c4 ; ¼ a0 þ pffiffiffiffiffi 1 1 e c2 1 þ tanh 2 c2 x þ c2 1 t pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2eb1 c2 c4 tan 12 c2 x þ 4c211 t þ b1 c3 b1 2e c2 c4 tan 12 c2 x þ 4c211 t þ c3 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi þ c2 < 0; ¼ a0 c2 sec2 12 c2 x þ 4c211 t c3 c2 sec4 12 c2 x þ 4c211 t pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2eb1 c2 c4 tanh 12 c2 x þ 4c211 t b1 c3 b1 ð2e c2 c4 tanh 12 c2 x þ 4c211 t c3 Þ2 pffiffiffiffiffi pffiffiffiffiffi ¼ a0 þ þ c2 > 0; 2 4 c2 sech 12 c2 x þ 4c211 t c3 c2 sech 12 c2 x þ 4c211 t pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2b1 c4 4b1 c4 pffiffiffiffiffi þ ¼ a0 þ pffiffiffiffiffi pffiffiffiffiffi 2 c2 > 0; c3 ¼ 2e c2 c4 ; 1 1 e c2 1 þ tanh 12 c2 x þ 4c211 t c3 1 þ tanh 2 c2 x þ 4c2 1 t
when a þ b ¼ 0; and
u77 ¼
2 1 þ c þ cc2 6c2 1 pffiffiffiffiffi 3c2 1 pffiffiffiffiffi 1 þ tanh 1 þ tanh c2 ðx þ ctÞ c2 ðx þ ctÞ 2 2 ca þ cb aþb aþb
c2 > 0;
pffiffiffiffiffiffiffiffiffi c3 ¼ 2e c2 c4
when a þ b – 0. Case 8. c0 ¼ c1 ¼ c2 ¼ 0; c4 < 0
a ¼ b; a1 ¼ 0; a2 ¼ 0; b2 ¼ 0; c ¼ 1: From (A.16) and (3.4), we find an exponential type solution when a þ b ¼ 0 as follow:
u81 ¼ a0 þ
2b1 c4 ec3 exp pffiffiffiffiffiffiffiffiffi ðx tÞ : c3 2 c4
Remark 1. As m ! 1 and m ! 0, the Jacobi elliptic doubly periodic type solutions u27 u215 can be reduced to u21 u23 through proper transformation. The solutions u21 u26 and u17 have been found by Elwakil [12] (by proper transformation). The other solutions we have found are all new. We also find the solutions when a þ b ¼ 0, which was not given by Elwakil. Remark 2. For illustration, the solutions for the GSWW equation are displayed in Figs. 1–3 for the cases when a þ b ¼ 0 and Figs. 4–7 for the cases when a þ b – 0.
Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
Fig. 1. Hyperbolic type solution u11 with a0 ¼ 1; b1 ¼ 1; c2 ¼ 2; c4 ¼ 1, and u64 with a0 ¼ 1; a1 ¼ 1; c1 ¼ 1; c2 ¼ 2.
Fig. 2. Exponential type solution u52 with a0 ¼ 2; a1 ¼ 1; c1 ¼ 1; c2 ¼ 1, and u76 with a0 ¼ 1; b1 ¼ 1; c2 ¼ 2; c4 ¼ 1.
Fig. 3. Triangular type solution u12 with a0 ¼ 1; b1 ¼ 1; c2 ¼ 1; c4 ¼ 1 and polynomial type solution u14 with a0 ¼ 1; b1 ¼ 1; b2 ¼ 1; c4 ¼ 1.
Fig. 4. Solitary wave solution u21 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1, and u23 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 2.
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Fig. 5. Solitary wave solution u53 with a ¼ 0:5; b ¼ 0:5; c0 ¼ 1; c1 ¼ 1; c2 ¼ 1; and u66 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1.
Fig. 6. Solitary wave solution u77 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1; and triangular type solution u22 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1.
Fig. 7. Jocobi elliptic doubly periodic type solution u213 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c2 ¼ 1=9; m ¼ 0:8; and Weierstass elliptic doubly periodic type solution u42 with a ¼ 0:5; b ¼ 0:5; c ¼ 1; c0 ¼ 0:1; c1 ¼ 0:2; c2 ¼ 0:1.
4. Conclusion In this paper we use a rational expansion to improve the Fan’s method for solving the GSWW equation. We successfully obtain kinds of new exact solutions of the GSWW equation by the proposed method. The new solutions, for examples, kink shaped solitary solution u77 , Jacobi elliptic doubly periodic type solution u27 , Weierstrass elliptic doubly periodic type solution u41 , exponential type solution u53 and triangular type solution u65 have never been found before. More nonlinear PDEs can be expected to be solved by this method. Acknowledgements The work described in this paper was fully supported by a Grant from CityU (Project No. 7001791).
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Appendix A The fundamental solutions of x0 ¼ e follows:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c0 þ c1 x þ c2 x2 þ c3 x3 þ c4 x4 for different choices of c0 ; c1 ; c2 ; c3 are listed as
Case 1. c0 ¼ c1 ¼ c3 ¼ 0 A bell shaped solitary wave solution, a triangular type solution and a rational solution
rffiffiffiffiffiffiffiffiffi pffiffiffiffiffi c x ¼ 2 sechð c2 nÞ; c2 > 0; c4 < 0; c4 rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi c x ¼ 2 sec ð c2 nÞ; c2 < 0; c4 > 0; c4
e x ¼ pffiffiffiffiffi ; c2 ¼ 0; c4 > 0: c4 n
ðA:1Þ ðA:2Þ ðA:3Þ
Case 2. c1 ¼ c3 ¼ 0 A kink shaped solitary wave solution, a triangular type solution and three Jacobi elliptic doubly periodic type solutions
rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi c2 c2 c2 tanh n ; c2 < 0; c4 > 0; c0 ¼ 2 ; 2c4 2 4c4 rffiffiffiffiffiffiffiffi rffiffiffiffiffi c c2 c2 n ; c2 > 0; c4 > 0; c0 ¼ 2 ; x ¼ e 2 tan 2c4 2 4c4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c2 m c2 c2 m2 ð1 m2 Þ n ; c2 > 0; c4 < 0; c0 ¼ 2 x¼ ; cn c4 ð2m2 1Þ 2m2 1 c4 ð2m2 1Þ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 c2 c2 ð1 m2 Þ dn x¼ ; n ; c2 > 0; c4 < 0; c0 ¼ 2 2 2 c4 ð2 m Þ 2m c4 ð2 m2 Þ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 m2 c2 c22 m2 n ; c2 < 0; c4 > 0; c0 ¼ x¼e ; sn 2 2 c4 ðm þ 1Þ m þ1 c4 ðm2 þ 1Þ2
x¼e
ðA:4Þ ðA:5Þ ðA:6Þ ðA:7Þ ðA:8Þ
where m is a modulus. Case 3. c0 ¼ c1 ¼ c4 ¼ 0 A bell shaped wave solitary solution, a triangular type solution and a rational type solution
pffiffiffiffiffi c2 c2 2 sech n ; c2 > 0; 2 c3 pffiffiffiffiffiffiffiffiffi c c2 x ¼ 2 sec2 n ; c2 < 0; c3 2 1 x ¼ 2 ; c2 ¼ 0: c3 n
x¼
ðA:9Þ ðA:10Þ ðA:11Þ
Case 4. c2 ¼ c4 ¼ 0; c0 – 0; c1 – 0; c3 > 0 A Weierstrass elliptic doubly periodic type solution
pffiffiffiffiffi c3 n; g 2 ; g 3 ; 2
x¼}
ðA:12Þ
where g 2 ¼ 4c1 =c3 and g 3 ¼ 4c0 =c3 are called invariants of Weierstrass elliptic function. Case 5. c2 ¼ c3 ¼ c4 ¼ 0 Two polynomial type solutions
pffiffiffiffiffi
x ¼ e c0 n; c1 ¼ 0; c0 > 0;
ðA:13Þ
c 1 x ¼ 0 þ c 1 n2 ; c1 4
ðA:14Þ
c1 – 0:
1994
Z. Yang, Y.C. Hon / Applied Mathematics and Computation 216 (2010) 1984–1995
Case 6. c0 ¼ c1 ¼ c2 ¼ 0 A rational type solution and a exponential type solution
4c3 ; c4 – 0; c23 n2 4c4 c ec3 ffin ; x ¼ 3 exp pffiffiffiffiffiffiffiffi 2c4 2 c4
x¼
ðA:15Þ c4 < 0:
ðA:16Þ
Case 7. c3 ¼ c4 ¼ 0 A exponential type solution, a triangular type solution and a hyperbolic type solution
pffiffiffiffiffi c1 c2 þ exp ðe c2 nÞ; c2 > 0; c0 ¼ 1 ; 2c2 4c2 pffiffiffiffiffiffiffiffiffi c ec x ¼ 1 þ 1 sin ð c2 nÞ; c0 ¼ 0; c2 < 0; 2c2 2c2 pffiffiffiffiffi c ec x ¼ 1 þ 1 sinh ð2 c2 nÞ; c0 ¼ 0; c2 > 0 2c2 2c2 rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi c x ¼ e 0 sin ð c2 nÞ; c1 ¼ 0; c0 > 0; c2 < 0; c2 rffiffiffiffiffi pffiffiffiffiffi c x ¼ e 0 sinh ð c2 nÞ; c1 ¼ 0; c0 > 0; c2 > 0: c2
x¼
ðA:17Þ ðA:18Þ ðA:19Þ ðA:20Þ ðA:21Þ
Case 8. c0 ¼ c1 ¼ 0; c4 > 0 A triangular type solution, two solitary wave solutions
pffiffiffiffiffiffiffiffiffi c2 sec2 12 c2 n pffiffiffiffiffiffiffiffiffi x ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; c2 < 0; 2e c2 c4 tan 12 c2 n þ c3 p ffiffiffiffi ffi 2 pffiffiffiffiffiffiffiffiffi c2 sech ð12 c2 nÞ pffiffiffiffiffi x ¼ pffiffiffiffiffiffiffiffiffi ; c2 > 0; c3 – 2e c2 c4 ; 2e c2 c4 tanh 12 c2 n c3 rffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 1 c 1 pffiffiffiffiffi x ¼ e 2 1 þ tanh c2 n ; c2 > 0; c3 ¼ 2e c2 c4 : 2 2 c4
ðA:22Þ ðA:23Þ ðA:24Þ
The Jacobi elliptic functions are doubly periodical and possess properties of triangular functions: 2
sn2 n þ cn2 n ¼ 1;
dn n ¼ 1 m2 sn2 n;
ðsnnÞ0 ¼ cnndnn;
ðcnnÞ0 ¼ snndnn;
ðdnnÞ0 ¼ m2 snncnn:
When m ! 1, the Jacobi functions degenerate to the following hyperbolic functions:
snn ! tanh n;
cnn ! sech n;
dnn ! sech n
and when m ! 0, the Jacobi functions degenerate to the following triangular functions:
snn ! sin n;
cnn ! cos n;
dnn ! 1:
A more detail notations for the Weierstrass and Jacobi elliptic functions can be found in [35,36]. When m ! 1, the Jacobi doubly periodic solutions (A.6) and (A.7) degenerate to the solitary wave solutions (A.1) and the solution (A.8) degenerates to (A.4). References [1] A. Espinosa, J. Fujioka, Hydrodynamic foundation and Pailevé analysis of the Hirota–Satsuma-type equations, J. Phys. Soc. Jpn. 63 (1994) 1289–1294. [2] P.A. Clarkson, E.L. Mansfield, On a shallow water wave equation, Nonlinearity 7 (1994) 975–1000. [3] M.J. Ablowitz, D.J. Kaup, A.C. Newell, Segur, The inverse scattering transform: fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974) 249– 315. [4] R. Hirota, J. Satsuma, N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Jpn. 40 (1976) 611–612. [5] P.A. Clarkson, E.L. Mansfield, Symmetry reductions and exact solutions of shallow water wave equations, Acta Appl. Math. 39 (1995) 245–276. [6] J. Weiss, M. Tabor, G. Carnevale, The Pailevé property for partial differential equations, J. Math. Phys. 24 (1983) 522–526. [7] M.J. Ablowitz, A. Ramani, H. Segur, A connection between nonlinear evolution equations and ordinary differential equations of P-type, I, J. Math. Phys. 21 (1980) 715–721.
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