- expansion method and its applications to the Broer – Kaup equations and approximate long water wave equations

By introducing a new general ansatze, the improved G^'G-expansion method is proposed to construct exact solutions of both Broer-Kaup equations and approximate long water wave equations. As a result, some new travelling wave solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions, are obtained. When the parameters are taken as special values, the solitary wave solutions are derived from the hyperbolic function solutions. The proposed method is straightforward, concise and effective, and can be applied to other nonlinear evolution equations in mathematical physics.

[1]  Deng-Shan Wang,et al.  Symbolic computation and non-travelling wave solutions of (2 + 1)-dimensional nonlinear evolution equations , 2008 .

[2]  Shimin Guo,et al.  Auxiliary equation method for the mKdV equation with variable coefficients , 2010, Appl. Math. Comput..

[3]  Abdul-Majid Wazwaz,et al.  The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations , 2007, Appl. Math. Comput..

[4]  Jike Wang,et al.  The novel solutions of auxiliary equation and their application to the (2 + 1)-dimensional Burgers equations , 2008 .

[5]  Shimin Guo,et al.  The extended (G'/G)-expansion method and its applications to the Whitham-Broer-Kaup-Like equations and coupled Hirota-Satsuma KdV equations , 2010, Appl. Math. Comput..

[6]  Xiangzheng Li,et al.  A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms , 2007 .

[7]  D. J. Kaup,et al.  A Higher-Order Water-Wave Equation and the Method for Solving It , 1975 .

[8]  Mingliang Wang,et al.  Periodic wave solutions to a coupled KdV equations with variable coefficients , 2003 .

[9]  A. Jamiołkowski.  Book reviewApplications of Lie groups to differential equations : Peter J. Olver (School of Mathematics, University of Minnesota, Minneapolis, U.S.A): Graduate Texts in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, XXVI+497pp. , 1989 .

[10]  Abdul-Majid Wazwaz,et al.  The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations , 2007, Appl. Math. Comput..

[11]  Abdul-Majid Wazwaz,et al.  The sine-cosine and the tanh methods: Reliable tools for analytic treatment of nonlinear dispersive equations , 2006, Appl. Math. Comput..

[12]  Jinliang Zhang,et al.  Application of the (G'/G)-expansion to travelling wave solutions of the Broer-Kaup and the approximate long water wave equations , 2008, Appl. Math. Comput..

[13]  C. Dai,et al.  Periodic structures based on variable separation solution of the (2+1)-dimensional Boiti–Leon–Pempinelli equation , 2009 .

[14]  Michael Tabor,et al.  Similarity reductions from extended Painleve´ expansions for nonintegrable evolution equations , 1991 .

[15]  Mingliang Wang,et al.  Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics , 1996 .

[16]  C. Dai,et al.  Semifoldons with fusion and fission properties of (2 + 1)-dimensional nonlinear system , 2008 .

[17]  Xiangzheng Li,et al.  Various exact solutions of nonlinear Schrödinger equation with two nonlinear terms , 2007 .

[18]  Wei Wang,et al.  A generalized (G′G)-expansion method for the mKdV equation with variable coefficients , 2008 .

[19]  Zhou Yu-bin,et al.  Application of Modified G′/G-Expansion Method to Traveling Wave Solutions for Whitham–Broer–Kaup-Like Equations , 2009 .

[20]  Abdul-Majid Wazwaz,et al.  The sine-cosine method for obtaining solutions with compact and noncompact structures , 2004, Appl. Math. Comput..

[21]  Mingliang Wang,et al.  The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations , 2004 .

[22]  Wen-Xiu Ma,et al.  Complexiton solutions to the Korteweg–de Vries equation , 2002 .

[23]  Mingliang Wang,et al.  The periodic wave solutions for the Klein–Gordon–Schrödinger equations , 2003 .

[24]  Mingliang Wang,et al.  The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , 2008 .

[25]  Ljf Lambert Broer.  Approximate equations for long water waves , 1975 .

[26]  Abdul-Majid Wazwaz,et al.  Multiple kink solutions and multiple singular kink solutions for the (2 + 1)-dimensional Burgers equations , 2008, Appl. Math. Comput..

[27]  Sheng Zhang.  A further improved extended Fan sub-equation method for (2+1)-dimensional breaking soliton equations , 2008, Appl. Math. Comput..

[28]  Xiangzheng Li,et al.  Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation , 2007 .

[29]  Mingliang Wang.  Exact solutions for a compound KdV-Burgers equation , 1996 .

[30]  Abdul-Majid Wazwaz,et al.  The extended tanh method for abundant solitary wave solutions of nonlinear wave equations , 2007, Appl. Math. Comput..

[31]  Mingliang Wang.  SOLITARY WAVE SOLUTIONS FOR VARIANT BOUSSINESQ EQUATIONS , 1995 .

[32]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[33]  G. B. Whitham,et al.  Variational methods and applications to water waves , 1967, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[34]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

[35]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .