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Exact solutions of (2+1) -dimensional Bogoyavlenskii’s breaking soliton equation with symbolic computation
Computers and Mathematics with Applications 60 (2010) 919–923
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Exact solutions of (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation with symbolic computation Tiecheng Xia ∗ , Shouquan Xiong Department of Mathematics, Shanghai University, Shanghai, 200444, PR China Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080, PR China
article
info
Article history: Received 8 May 2009 Received in revised form 22 May 2010 Accepted 24 May 2010 Keywords: (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation 0 ( GG )-expansion method Hyperbolic function solutions Trigonometric function solutions Rational solutions
abstract 0
In this paper, the generalized ( GG )-expansion method (Wang et al. (2008) and Zhang et al. (2008) [13,14]) is used to construct exact solutions of the (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation. As a result, non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. If the parameters take different values, some 0 more solutions are derived. It is shown that the generalized ( GG )-expansion method, with the help of symbolic computation, provides an effective and powerful method for solving high-dimensional nonlinear partial differential equations in mathematical physics. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction It is well known that nonlinear partial differential equations (NLPDEs) are widely used to describe complex phenomena in various fields of science, such as hydrodynamics, solid-state physics, plasma physics. The study of exact solutions of these nonlinear equations plays an important role in soliton theory. Thus, a number of powerful methods have been presented. For example, the inverse scattering method [1], Hirota’s bilinear method [2], Bäcklund transformation [3], the homogeneous balance method [4], variational method [5,6], algebra method [7], sine–cosine method [8], the Jacobi elliptic function method [9], the F-expansion method [10] and so on. The (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation ut + uxxy + 4uuy + 4ux ∂ −1 uy = 0,
(1.1)
its equivalent form [11] ut + uxxy + 4uvx + 4ux v = 0,
(1.2)
uy = v x ,
(1.3)
which describes the (2 + 1)-dimensional interaction of a Riemann wave propagating along the y axis with a long wave long the x axis. The u = u(x, y, t ) represents the physical field and v = v(x, y, t ) some potential. This equation is typical of the so-called ‘‘breaking soliton’’ equation and was studied by Bogoyavenskii, where overlapping solutions were generated [12].
∗ Corresponding author at: Department of Mathematics, Shanghai University, Shanghai, 200444, PR China. Tel.: +86 021 66132511; fax: +86 021 66133992. E-mail address: [email protected] (T. Xia). 0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.05.037
920
T. Xia, S. Xiong / Computers and Mathematics with Applications 60 (2010) 919–923
In this paper, we apply the generalized
0 G G
-expansion method to the (2 + 1)-dimensional Bogoyavlenskii’s breaking
soliton equation with variable coefficients, as a result, hyperbolic function solutions, trigonometric function solutions and rational solutions with parameters are obtained. The rest of this paper is organized as follows: In the next section, we give the description of the generalized
G0 G
-
expansion method with variable coefficients. In Section 3, we apply this method to the (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation. In the last section, some conclusions are given. 2. The description of the method For a given NLPDEs with independent variables X = (x, y, z , . . . , t ) and dependent variable u:
Φ (u, ux , uy , ut , uxx , uxy , uxt , . . .) = 0.
(2.1)
We seek its solutions in this general form: u=
n X
ai (Y )
i =0
G0 (ξ )
i (2.2)
G(ξ )
where Y = (y, z , . . . , t ), G = G(ξ ), ξ = ξ (X ) and ai (Y ) (i = 0, . . . , n) to be determined later, and G satisfies: G00 + λG + µG = 0.
(2.3)
To determine u explicitly, we take the following four steps: Step 1: Determine the degree n by balancing the highest degree and nonlinear term in Eq. (2.1). Step 2: Substituting Eqs. (2.2) and (2.3) into Eq. (2.1), yields a polynomial of
0 G G
. Setting the coefficients of
0 i G G
to zero
lead out a set of over-determined nonlinear partial differential equations of ai (Y ) and ξ . Step 3: Solving the over-determined nonlinear partial differential equations that we have got in Step 2 with the help of Maple, we can work out the explicit expressions of ai (Y ) and ξ . Step 4: Substituting the results obtained in Step 3 and the well-known solutions of Eq. (2.3) into (2.2), then we can obtain exact solutions of Eq. (2.1). 3. Application to the (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation According to the method, by balancing the highest nonlinear term and the highest order derivative term, the ansatz solutions of Eqs. (1.2) and (1.3) can be written as the following form: u = a2 (y, t )
v = b2 (y, t )
G0 (ξ )
2
G(ξ )
G0 (ξ ) G(ξ )
2
+ a1 (y, t ) + b1 (y, t )
G0 (ξ )
G(ξ )
G0 (ξ ) G(ξ )
+ a0 (y, t )
(3.1)
+ b0 (y, t )
(3.2)
where ξ = kx + η, η = η(y, t ) and a2 b2 6= 0. With the aid of Maple, substituting ansatz solutions (3.1) and (3.2) along with Eq. (2.3) into Eqs. (1.2) and (1.3), the left
0 i
(i = 0, 1, 2, . . .). Setting each coefficient of to zero yields a set of over-determined nonlinear partial differential equations of ai (y, t ), bi (y, t ) and η as follows: 0 0 G : a0t + a1y µλk2 + 2a2y µ2 k2 − a1 (λ2 − µ)µηy k2 − 4a0 b1 µk − 3a1 µ2 k2 ηy − a1 µηt sides of Eqs. (1.2) and (1.3) are converted into finite power series of
G G
G
− 6a2 µ2 k2 ληy − 4a1 µkb0 = 0, 0 1 G : 6a2y λµk2 − 24a2 µ2 k2 ηy − 2a2 µηt − 10a1 µλk2 ηy − 6(a2 λ2 k2 µ + a2 µk2 (λ2 − µ))ηy G
+ 3a1y µk2 − 8a1 b1 µk − 4a1 λkb0 + a1y (λ2 − µ)k2 + a1 (−λ3 + 2µλ)ηy k2 − 8a0 b2 µk
− 4a0 b1 λk − 2a2 (λ2 − µ)µηy k2 + a1t − 8a2 µkb0 − a1 ληt = 0, 2 G0 : 2a2y λ2 k2 − a1 ηt − 4a1 (λ2 − µ)k2 ηy − 8a1 b1 λk + 2a2y (λ2 − µ)k2 G
+ 2a2 (−λ3 + 2µλ)ηy k2 + a2t − 62a2 λµk2 ηy − 2a2 ληt − 6a2 λk2 (λ2 − µ)ηy
0 i G G
T. Xia, S. Xiong / Computers and Mathematics with Applications 60 (2010) 919–923
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+ 3a1y λk2 − 4a0 b1 k − 3a1 λ2 k2 ηy − 12a2 b1 µk + 10a2y µk2 − 12a1 b2 µk
− 4a1 kb0 − 8a2 λkb0 − 12a1 µk2 ηy − 8a0 b2 λk = 0, 3 G0 : −16a2 b2 µk + 2a1y k2 − 12a2 b1 λk − 8a2 kb0 − 14a2 (λ2 − µ)ηy k2 − 8a1 b1 k − 24a2 λ2 k2 ηy G
− 2a2 ηt − 54a2 µk2 ηy − 8a0 b2 k − 12a1 b2 λk + 10a2y λk2 − 12a1 λk2 ηy = 0, 4 G0 : −16a2 b2 λk − 12a2 b1 k − 6a1 k2 ηy − 54a2 λk2 ηy − 12a1 b2 k + 6a2y k2 = 0, G
G0
5
: −16a2 b2 k − 24a2 k2 ηy = 0,
G
G0
0
: −a1 µk + b1 µηy − b0y = 0,
G
G0
1
: 2b2 µηy − a1 λk − b1y + b1 ληy − 2a2 µk = 0,
G
G0
2
: −2a2 λk + 2b2 ληy − a1 k − b2y + b1 ηy = 0,
G
G0
3
: 2b2 ηy − 2a2 k = 0.
G
Solving the above over-determined nonlinear partial differential equations by using Maple, we have: a0 =
1 4
2
2
(−2µf 0 (y) − λ2 f 0 (y) + 3λf 00 (y) − f 000 (y)/f 0 (y)), 1 b0 = − (6µkf 0 (y) + g 0 (t )), 4
3 2 a2 = − f 0 (y), 2 3 b2 = − kf 0 (y), 2
a1 =
3 2
2
(f 00 (y) − f 0 (y)λ),
3 b1 = − kλf 0 (y), 2
η = kx + f (y) + g (t ),
(3.3) (3.4) (3.5)
where f (y) and g (t ) are arbitrary functions of the indicated variables, g 0 (t ) = dg (t )/dt and f 0 (y) = df (y)/dy. Substituting (3.3)–(3.5) into (3.1) and (3.2), we have: G0 (ξ ) 3 G0 (ξ ) 3 2 2 + (f 00 (y) − f 0 (y)λ) u = − f 0 (y) 2 G(ξ ) 2 G(ξ ) 1 2 02 00 000 02 + (−2µf (y) − λ f (y) + 3λf (y) − f (y)/f 0 (y)), 4
3
v = − kf 0 (y) 2
G0 (ξ )
2
G(ξ )
3
− kλf 0 (y)
2
G0 (ξ ) G(ξ )
(3.6)
1
− (6µkf 0 (y) + g 0 (t )) 4
(3.7)
where ξ = kx + f (y) + g (t ). Substituting the general solutions of Eq. (2.3) into (3.6) and (3.7), three types exact solutions of Eqs. (1.2) and (1.3) as follows: When λ2 − 4µ > 0, we get hyperbolic function solutions:
3 2 u1 = − f 0 (y) 2
√
λ2 −4µ
C1 sinh
2
√
λ 2 −4 µ
ξ
√
2
ξ
√
λ2 −4µ
2
+ C2 sinh ξ 2 √ √ λ2 −4µ λ2 −4µ C1 sinh ξ + C cosh ξ 2 2 2 3 2 √ √ + (f 00 (y) − f 0 (y)λ) 2 λ2 −4µ λ2 −4µ C1 cosh ξ + C2 sinh ξ 2 2 1
C1 cosh
2
2
2
ξ
+ C2 cosh
λ2 −4µ
+ (−2µf 0 (y) − λ2 f 0 (y) + 3λf 00 (y) − f 000 (y)/f 0 (y)), 4
(3.8)
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T. Xia, S. Xiong / Computers and Mathematics with Applications 60 (2010) 919–923
3 v1 = − kf 0 (y) 2
√
λ2 −4µ
C1 sinh
2
√ C1 cosh
3 − kλf 0 (y) 2
λ2 −4µ 2
ξ
ξ
2
λ2 −4µ
C1 cosh
2
2
√
λ2 −4µ
√
λ2 −4µ
+ C2 cosh
√ C1 sinh
√
λ2 −4µ
+ C2 sinh ξ
2
√
ξ
+ C2 cosh
ξ
2
λ2 −4µ 2
√
ξ
+ C2 sinh
λ2 −4µ 2
ξ ξ
1 0 0 − (6µkf (y) + g (t )) 4
(3.9)
where ξ = kx + f (y) + g (t ), C1 and C2 are arbitrary constants. When λ2 − 4µ < 0, we get trigonometric function solutions:
√
λ2 −4µ
√
λ2 −4µ
2
ξ + C2 cos ξ 2 √ 2 2 λ −4µ λ −4µ C1 cos ξ + C2 sin ξ 2 2 √ √ λ2 −4µ λ2 −4µ −C1 sin ξ + C cos ξ 2 2 2 3 2 √ √ + (f 00 (y) − f 0 (y)λ) 2 λ2 −4µ λ2 −4µ ξ + C2 sin ξ C1 cos 2 2
3 2 u2 = − f 0 (y) 2
1
−C1 sin √
2
2
2
+ (−2µf 0 (y) − λ2 f 0 (y) + 3λf 00 (y) − f 000 (y)/f 0 (y)),
(3.10)
4
√
λ2 −4µ
√
λ2 −4µ
2
ξ + C2 cos ξ 2 √ 2 2 2 λ −4µ λ −4 µ ξ + C2 sin ξ C1 cos 2 2 √ √ λ2 −4µ λ2 −4µ ξ + C cos ξ −C1 sin 2 2 2 1 3 0 0 √ √ − kλf 0 (y) − 4 (6µkf (y) + g (t )) 2 λ2 −4µ λ2 −4µ C1 cos ξ + C2 sin ξ 2 2
3 v2 = − kf 0 (y)
−C1 sin √
2
(3.11)
where ξ = kx + f (y) + g (t ), C1 and C2 are arbitrary constants. When λ2 − 4µ = 0, we get rational solutions: 3 2 u3 = − f 0 (y) 2 3
v3 = − kf 0 (y) 2
C2
2
C1 + C2 ξ
C2 C1 + C2 ξ
3
+ f 00 (y) 2
2
C2
C1 + C2 ξ
1
2
+ (−2µf 0 (y) − f 000 (y)/f 0 (y)), 4
1
− (6µkf 0 (y) + g 0 (t ))
(3.12)
(3.13)
4
where ξ = kx + f (y) + g (t ), C1 and C2 are arbitrary constants. Remark. All solutions in this letter have been checked with Maple by putting them back into the original Eqs. (1.2) and (1.3). 4. Conclusions With the help of symbolic computation, applying the generalized
0 G G
-expansion method to the (2 + 1)-dimensional
Bogoyavlenskii’s breaking soliton equation, more and new non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. If the parameters take different values, different hyperbolic function solutions, trigonometric function solutions and rational solutions are derived. To our knowledge, these new solutions have not been reported in former literature, they may be of significant importance for the explanation of some special physical phenomena. This method can be applied to high-dimensional NLPDEs in mathematical physics.
T. Xia, S. Xiong / Computers and Mathematics with Applications 60 (2010) 919–923
923
Acknowledgements We would like to express our sincere thanks to referee for the valuable advice. This project was supported by the Natural Science Foundation of Shanghai (Grant No. 09 ZR1410800), the Science Foundation of Key Laboratory of Mathematics Mechanization (Grant No. KLMM0806), the Shanghai Leading Academic Discipline Project (Grant No. J50101) and by Key Disciplines of Shanghai Municipality (S30104). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Solution Equations and Inverse Scatting, Cambridge University Press, New York, 1991. R. Hirota, Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971) 1192–1194. M.R. Miurs, Bäcklund Transformation, Springer, Berlin, 1978. M.L. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A 199 (1995) 169–172. J.H. He, Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fractals 19 (2004) 847–851. J.H. He, Variational approach to (2 + 1)-dimensional dispersive long water equations, Phys. Lett. A 335 (2005) 182–184. J.Q. Hu, An algebraic method exactly solving two high-dimensional nonlinear evolution equations, Chaos Solitons Fractals 23 (2005) 391–398. C.T. Yan, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77–84. S.K. Liu, Z.T. Fu, S.D. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289 (2001) 69–74. [10] Y.B. Zhou, M.L. Wang, Y.M. Wang, Periodic wave solutions to a coupled KdV equations with variable coefficients, Phys. Lett. A 308 (2003) 31–36. [11] E.G. Fan, Y.C. Hon, Quasiperiodic waves and asymptotic behavior for Bogoyavlenskii’s breaking soliton equation in (2 + 1) dimensions, Phys. Rev. E 78 (2008) 036607. [12] O.I. Bogoyavlenskii, Breaking solitons in (2 + 1)-dimensional integrable equations, Russian Math. Surveys 45 (1990) 1–86. [13] M.L. Wang, X.Z. Li, J.L. Zhang, The
G0 G
-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,
Phys. Lett. A 372 (2008) 417–423. [14] S. Zhang, J.L. Tong, W. Wang, A generalized 2254–2257.
0 G G
-expansion method for the mKdV equation with variable coefficients, Phys. Lett. A 372 (2008)
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Exact solutions of (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation with symbolic computation Tiecheng Xia ∗ , Shouquan Xiong Department of Mathematics, Shanghai University, Shanghai, 200444, PR China Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080, PR China
article
info
Article history: Received 8 May 2009 Received in revised form 22 May 2010 Accepted 24 May 2010 Keywords: (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation 0 ( GG )-expansion method Hyperbolic function solutions Trigonometric function solutions Rational solutions
abstract 0
In this paper, the generalized ( GG )-expansion method (Wang et al. (2008) and Zhang et al. (2008) [13,14]) is used to construct exact solutions of the (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation. As a result, non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. If the parameters take different values, some 0 more solutions are derived. It is shown that the generalized ( GG )-expansion method, with the help of symbolic computation, provides an effective and powerful method for solving high-dimensional nonlinear partial differential equations in mathematical physics. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction It is well known that nonlinear partial differential equations (NLPDEs) are widely used to describe complex phenomena in various fields of science, such as hydrodynamics, solid-state physics, plasma physics. The study of exact solutions of these nonlinear equations plays an important role in soliton theory. Thus, a number of powerful methods have been presented. For example, the inverse scattering method [1], Hirota’s bilinear method [2], Bäcklund transformation [3], the homogeneous balance method [4], variational method [5,6], algebra method [7], sine–cosine method [8], the Jacobi elliptic function method [9], the F-expansion method [10] and so on. The (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation ut + uxxy + 4uuy + 4ux ∂ −1 uy = 0,
(1.1)
its equivalent form [11] ut + uxxy + 4uvx + 4ux v = 0,
(1.2)
uy = v x ,
(1.3)
which describes the (2 + 1)-dimensional interaction of a Riemann wave propagating along the y axis with a long wave long the x axis. The u = u(x, y, t ) represents the physical field and v = v(x, y, t ) some potential. This equation is typical of the so-called ‘‘breaking soliton’’ equation and was studied by Bogoyavenskii, where overlapping solutions were generated [12].
∗ Corresponding author at: Department of Mathematics, Shanghai University, Shanghai, 200444, PR China. Tel.: +86 021 66132511; fax: +86 021 66133992. E-mail address: [email protected] (T. Xia). 0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.05.037
920
T. Xia, S. Xiong / Computers and Mathematics with Applications 60 (2010) 919–923
In this paper, we apply the generalized
0 G G
-expansion method to the (2 + 1)-dimensional Bogoyavlenskii’s breaking
soliton equation with variable coefficients, as a result, hyperbolic function solutions, trigonometric function solutions and rational solutions with parameters are obtained. The rest of this paper is organized as follows: In the next section, we give the description of the generalized
G0 G
-
expansion method with variable coefficients. In Section 3, we apply this method to the (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation. In the last section, some conclusions are given. 2. The description of the method For a given NLPDEs with independent variables X = (x, y, z , . . . , t ) and dependent variable u:
Φ (u, ux , uy , ut , uxx , uxy , uxt , . . .) = 0.
(2.1)
We seek its solutions in this general form: u=
n X
ai (Y )
i =0
G0 (ξ )
i (2.2)
G(ξ )
where Y = (y, z , . . . , t ), G = G(ξ ), ξ = ξ (X ) and ai (Y ) (i = 0, . . . , n) to be determined later, and G satisfies: G00 + λG + µG = 0.
(2.3)
To determine u explicitly, we take the following four steps: Step 1: Determine the degree n by balancing the highest degree and nonlinear term in Eq. (2.1). Step 2: Substituting Eqs. (2.2) and (2.3) into Eq. (2.1), yields a polynomial of
0 G G
. Setting the coefficients of
0 i G G
to zero
lead out a set of over-determined nonlinear partial differential equations of ai (Y ) and ξ . Step 3: Solving the over-determined nonlinear partial differential equations that we have got in Step 2 with the help of Maple, we can work out the explicit expressions of ai (Y ) and ξ . Step 4: Substituting the results obtained in Step 3 and the well-known solutions of Eq. (2.3) into (2.2), then we can obtain exact solutions of Eq. (2.1). 3. Application to the (2 + 1)-dimensional Bogoyavlenskii’s breaking soliton equation According to the method, by balancing the highest nonlinear term and the highest order derivative term, the ansatz solutions of Eqs. (1.2) and (1.3) can be written as the following form: u = a2 (y, t )
v = b2 (y, t )
G0 (ξ )
2
G(ξ )
G0 (ξ ) G(ξ )
2
+ a1 (y, t ) + b1 (y, t )
G0 (ξ )
G(ξ )
G0 (ξ ) G(ξ )
+ a0 (y, t )
(3.1)
+ b0 (y, t )
(3.2)
where ξ = kx + η, η = η(y, t ) and a2 b2 6= 0. With the aid of Maple, substituting ansatz solutions (3.1) and (3.2) along with Eq. (2.3) into Eqs. (1.2) and (1.3), the left
0 i
(i = 0, 1, 2, . . .). Setting each coefficient of to zero yields a set of over-determined nonlinear partial differential equations of ai (y, t ), bi (y, t ) and η as follows: 0 0 G : a0t + a1y µλk2 + 2a2y µ2 k2 − a1 (λ2 − µ)µηy k2 − 4a0 b1 µk − 3a1 µ2 k2 ηy − a1 µηt sides of Eqs. (1.2) and (1.3) are converted into finite power series of
G G
G
− 6a2 µ2 k2 ληy − 4a1 µkb0 = 0, 0 1 G : 6a2y λµk2 − 24a2 µ2 k2 ηy − 2a2 µηt − 10a1 µλk2 ηy − 6(a2 λ2 k2 µ + a2 µk2 (λ2 − µ))ηy G
+ 3a1y µk2 − 8a1 b1 µk − 4a1 λkb0 + a1y (λ2 − µ)k2 + a1 (−λ3 + 2µλ)ηy k2 − 8a0 b2 µk
− 4a0 b1 λk − 2a2 (λ2 − µ)µηy k2 + a1t − 8a2 µkb0 − a1 ληt = 0, 2 G0 : 2a2y λ2 k2 − a1 ηt − 4a1 (λ2 − µ)k2 ηy − 8a1 b1 λk + 2a2y (λ2 − µ)k2 G
+ 2a2 (−λ3 + 2µλ)ηy k2 + a2t − 62a2 λµk2 ηy − 2a2 ληt − 6a2 λk2 (λ2 − µ)ηy
0 i G G
T. Xia, S. Xiong / Computers and Mathematics with Applications 60 (2010) 919–923
921
+ 3a1y λk2 − 4a0 b1 k − 3a1 λ2 k2 ηy − 12a2 b1 µk + 10a2y µk2 − 12a1 b2 µk
− 4a1 kb0 − 8a2 λkb0 − 12a1 µk2 ηy − 8a0 b2 λk = 0, 3 G0 : −16a2 b2 µk + 2a1y k2 − 12a2 b1 λk − 8a2 kb0 − 14a2 (λ2 − µ)ηy k2 − 8a1 b1 k − 24a2 λ2 k2 ηy G
− 2a2 ηt − 54a2 µk2 ηy − 8a0 b2 k − 12a1 b2 λk + 10a2y λk2 − 12a1 λk2 ηy = 0, 4 G0 : −16a2 b2 λk − 12a2 b1 k − 6a1 k2 ηy − 54a2 λk2 ηy − 12a1 b2 k + 6a2y k2 = 0, G
G0
5
: −16a2 b2 k − 24a2 k2 ηy = 0,
G
G0
0
: −a1 µk + b1 µηy − b0y = 0,
G
G0
1
: 2b2 µηy − a1 λk − b1y + b1 ληy − 2a2 µk = 0,
G
G0
2
: −2a2 λk + 2b2 ληy − a1 k − b2y + b1 ηy = 0,
G
G0
3
: 2b2 ηy − 2a2 k = 0.
G
Solving the above over-determined nonlinear partial differential equations by using Maple, we have: a0 =
1 4
2
2
(−2µf 0 (y) − λ2 f 0 (y) + 3λf 00 (y) − f 000 (y)/f 0 (y)), 1 b0 = − (6µkf 0 (y) + g 0 (t )), 4
3 2 a2 = − f 0 (y), 2 3 b2 = − kf 0 (y), 2
a1 =
3 2
2
(f 00 (y) − f 0 (y)λ),
3 b1 = − kλf 0 (y), 2
η = kx + f (y) + g (t ),
(3.3) (3.4) (3.5)
where f (y) and g (t ) are arbitrary functions of the indicated variables, g 0 (t ) = dg (t )/dt and f 0 (y) = df (y)/dy. Substituting (3.3)–(3.5) into (3.1) and (3.2), we have: G0 (ξ ) 3 G0 (ξ ) 3 2 2 + (f 00 (y) − f 0 (y)λ) u = − f 0 (y) 2 G(ξ ) 2 G(ξ ) 1 2 02 00 000 02 + (−2µf (y) − λ f (y) + 3λf (y) − f (y)/f 0 (y)), 4
3
v = − kf 0 (y) 2
G0 (ξ )
2
G(ξ )
3
− kλf 0 (y)
2
G0 (ξ ) G(ξ )
(3.6)
1
− (6µkf 0 (y) + g 0 (t )) 4
(3.7)
where ξ = kx + f (y) + g (t ). Substituting the general solutions of Eq. (2.3) into (3.6) and (3.7), three types exact solutions of Eqs. (1.2) and (1.3) as follows: When λ2 − 4µ > 0, we get hyperbolic function solutions:
3 2 u1 = − f 0 (y) 2
√
λ2 −4µ
C1 sinh
2
√
λ 2 −4 µ
ξ
√
2
ξ
√
λ2 −4µ
2
+ C2 sinh ξ 2 √ √ λ2 −4µ λ2 −4µ C1 sinh ξ + C cosh ξ 2 2 2 3 2 √ √ + (f 00 (y) − f 0 (y)λ) 2 λ2 −4µ λ2 −4µ C1 cosh ξ + C2 sinh ξ 2 2 1
C1 cosh
2
2
2
ξ
+ C2 cosh
λ2 −4µ
+ (−2µf 0 (y) − λ2 f 0 (y) + 3λf 00 (y) − f 000 (y)/f 0 (y)), 4
(3.8)
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T. Xia, S. Xiong / Computers and Mathematics with Applications 60 (2010) 919–923
3 v1 = − kf 0 (y) 2
√
λ2 −4µ
C1 sinh
2
√ C1 cosh
3 − kλf 0 (y) 2
λ2 −4µ 2
ξ
ξ
2
λ2 −4µ
C1 cosh
2
2
√
λ2 −4µ
√
λ2 −4µ
+ C2 cosh
√ C1 sinh
√
λ2 −4µ
+ C2 sinh ξ
2
√
ξ
+ C2 cosh
ξ
2
λ2 −4µ 2
√
ξ
+ C2 sinh
λ2 −4µ 2
ξ ξ
1 0 0 − (6µkf (y) + g (t )) 4
(3.9)
where ξ = kx + f (y) + g (t ), C1 and C2 are arbitrary constants. When λ2 − 4µ < 0, we get trigonometric function solutions:
√
λ2 −4µ
√
λ2 −4µ
2
ξ + C2 cos ξ 2 √ 2 2 λ −4µ λ −4µ C1 cos ξ + C2 sin ξ 2 2 √ √ λ2 −4µ λ2 −4µ −C1 sin ξ + C cos ξ 2 2 2 3 2 √ √ + (f 00 (y) − f 0 (y)λ) 2 λ2 −4µ λ2 −4µ ξ + C2 sin ξ C1 cos 2 2
3 2 u2 = − f 0 (y) 2
1
−C1 sin √
2
2
2
+ (−2µf 0 (y) − λ2 f 0 (y) + 3λf 00 (y) − f 000 (y)/f 0 (y)),
(3.10)
4
√
λ2 −4µ
√
λ2 −4µ
2
ξ + C2 cos ξ 2 √ 2 2 2 λ −4µ λ −4 µ ξ + C2 sin ξ C1 cos 2 2 √ √ λ2 −4µ λ2 −4µ ξ + C cos ξ −C1 sin 2 2 2 1 3 0 0 √ √ − kλf 0 (y) − 4 (6µkf (y) + g (t )) 2 λ2 −4µ λ2 −4µ C1 cos ξ + C2 sin ξ 2 2
3 v2 = − kf 0 (y)
−C1 sin √
2
(3.11)
where ξ = kx + f (y) + g (t ), C1 and C2 are arbitrary constants. When λ2 − 4µ = 0, we get rational solutions: 3 2 u3 = − f 0 (y) 2 3
v3 = − kf 0 (y) 2
C2
2
C1 + C2 ξ
C2 C1 + C2 ξ
3
+ f 00 (y) 2
2
C2
C1 + C2 ξ
1
2
+ (−2µf 0 (y) − f 000 (y)/f 0 (y)), 4
1
− (6µkf 0 (y) + g 0 (t ))
(3.12)
(3.13)
4
where ξ = kx + f (y) + g (t ), C1 and C2 are arbitrary constants. Remark. All solutions in this letter have been checked with Maple by putting them back into the original Eqs. (1.2) and (1.3). 4. Conclusions With the help of symbolic computation, applying the generalized
0 G G
-expansion method to the (2 + 1)-dimensional
Bogoyavlenskii’s breaking soliton equation, more and new non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. If the parameters take different values, different hyperbolic function solutions, trigonometric function solutions and rational solutions are derived. To our knowledge, these new solutions have not been reported in former literature, they may be of significant importance for the explanation of some special physical phenomena. This method can be applied to high-dimensional NLPDEs in mathematical physics.
T. Xia, S. Xiong / Computers and Mathematics with Applications 60 (2010) 919–923
923
Acknowledgements We would like to express our sincere thanks to referee for the valuable advice. This project was supported by the Natural Science Foundation of Shanghai (Grant No. 09 ZR1410800), the Science Foundation of Key Laboratory of Mathematics Mechanization (Grant No. KLMM0806), the Shanghai Leading Academic Discipline Project (Grant No. J50101) and by Key Disciplines of Shanghai Municipality (S30104). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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