An extended method and its application to Whitham-Broer-Kaup equation and two-dimensional perturbed KdV equation

Abstract An extended method is proposed for constructing new forms of exact travelling wave solutions to nonlinear evolution equations by making a more general transformation. For illustration, we apply the extended method to Whitham–Broer–Kaup equation and two-dimensional perturbed KdV equation and successfully cover the previously known travelling wave solutions found by Chen’s method [Chaos, Solitons and Fractals 22 (2004) 675; Int. J. Mod. Phys. C 4 (2004) 595]. Moreover, some new forms of solutions are obtained.

[1]  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.

[2]  W. Malfliet.  Solitary wave solutions of nonlinear wave equations , 1992 .

[3]  Andrew G. Glen,et al.  APPL , 2001 .

[4]  W. Hereman,et al.  The tanh method: I. Exact solutions of nonlinear evolution and wave equations , 1996 .

[5]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[6]  Zhuosheng Lü,et al.  On a further extended tanh method , 2003 .

[7]  Hong-qing Zhang,et al.  Soliton-like and period form solutions for high dimensional nonlinear evolution equations , 2003 .

[8]  Qi Wang,et al.  CONSTRUCTING FAMILIES TRAVELING WAVE SOLUTIONS IN TERMS OF SPECIAL FUNCTION FOR THE ASYMMETRIC NIZHNIK NOVIKOV VESSELOV EQUATION , 2004 .

[9]  Yong Chen,et al.  Explicit exact solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order , 2003 .

[10]  Leon M. Hall,et al.  Special Functions , 1998 .

[11]  Qi Wang,et al.  A generalized method and general form solutions to the Whitham–Broer–Kaup equation , 2004 .

[12]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[13]  Engui Fan,et al.  Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations , 2002 .

[14]  E. J. Parkes,et al.  Travelling solitary wave solutions to a compound KdV-Burgers equation , 1997 .

[15]  Zhenya Yan,et al.  New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water , 2001 .

[16]  Wen-Xiu Ma,et al.  THE BI-HAMILTONIAN STRUCTURE OF THE PERTURBATION EQUATIONS OF THE KDV HIERARCHY , 1996 .

[17]  B. Duffy,et al.  An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations , 1996 .

[18]  Engui Fan,et al.  An algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinea , 2003 .

[19]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .

[20]  Yong Chen,et al.  General projective Riccati equation method and exact solutions for generalized KdV-type and KdV–Burgers-type equations with nonlinear terms of any order , 2004 .

[21]  S. A. El-Wakil,et al.  Modified extended tanh-function method for solving nonlinear partial differential equations , 2002 .

[22]  E. Fan,et al.  Extended tanh-function method and its applications to nonlinear equations , 2000 .

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

[24]  Bo Tian,et al.  Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics , 2001 .

[25]  Zhenya Yan.  New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations , 2001 .

[26]  J. Rogers.  Chaos , 1876 .

[27]  Li Biao,et al.  Symbolic Computation and Construction of Soliton-Like Solutions to the (2+1)-Dimensional Breaking Soliton Equation , 2003 .

[28]  R. Hirota.  Exact solution of the Korteweg-deVries equation for multiple collision of solitons , 1971 .