New Exact Solutions of Some Nonlinear Partial Differential Equations

The modified extended tanh-function (METF) method for finding solitary wave solu- tions to nonlinear physical problems is described. We consider four kinds of non-linear partial differential equations such as modified Korteweg-de Vries equation (MKdV for short), two- dimensional Korteweg-de Vries equation (2DKdV-Burgers for short), variant nonlinear water wave equation and variant Boussinesq equations. Their solitary wave solutions are constructed as well. The method is usually tedious to use by hand, therefore we use the Maple package to solve the algebraic system and consequently to get the exact solutions.

[1]  Chuntao Yan.  A simple transformation for nonlinear waves , 1996 .

[2]  A. A. Soliman,et al.  The numerical simulation for stiff systems of ordinary differential equations , 2007, Comput. Math. Appl..

[3]  A. A. Soliman,et al.  Numerical solutions of nonlinear evolution equations using variational iteration method , 2007 .

[4]  A. A. Soliman,et al.  New applications of variational iteration method , 2005 .

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

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

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

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

[9]  D. Kaya.  An application for the higher order modified KdV equation by decomposition method , 2005 .

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

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

[12]  A. A. Soliman,et al.  The modified extended tanh-function method for solving Burgers-type equations , 2006 .

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

[14]  Dogan Kaya,et al.  The decomposition method for solving (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional KP equation , 2004, Appl. Math. Comput..

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

[16]  M. C. Shen,et al.  Asymptotic Theory of Unsteady Three-Dimensional Waves in a Channel of Arbitrary Cross Section , 1969 .

[17]  Wen-Xiu Ma,et al.  An exact solution to two-dimensional Korteweg-de Vries-Burgers equation , 1993 .

[18]  A. A. Soliman,et al.  Numerical simulation of the generalized regularized long wave equation by He's variational iteration method , 2005, Math. Comput. Simul..

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

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

[21]  Yan Zhen-ya,et al.  Auto-darboux transformation and exact solutions of the Brusselator reaction diffusion model , 2001 .

[22]  Abdou,et al.  VARIATIONAL ITERATION METHOD FOR SOLVING BURGERS AND COUPLED BURGERS EQUATIONS , 2005 .

[23]  Dogan Kaya.  Exact and numerical soliton solutions of some nonlinear physical models , 2004, Appl. Math. Comput..

[24]  M. Wadati,et al.  Relationships among Inverse Method, Bäcklund Transformation and an Infinite Number of Conservation Laws , 1975 .

[25]  A. A. Soliman,et al.  A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations , 2006 .

[26]  A. A. Soliman.  Collocation solution of the Korteweg–de Vries equation using septic splines , 2004, Int. J. Comput. Math..

[27]  Zuntao Fu,et al.  New transformations and new approach to find exact solutions to nonlinear equations , 2002 .

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

[29]  A. A. Soliman,et al.  Modified extended tanh-function method and its application on nonlinear physical equations , 2006 .

[30]  S. A. El-Wakil,et al.  Modified extended tanh-function method and its applications to nonlinear equations , 2005, Appl. Math. Comput..