Notes on the equivalence of different variable separation approaches for nonlinear evolution equations

Abstract The equivalence of multilinear variable separation approach, the extended projective Ricatti equation method and the improved tanh-function method is firstly reported when these three popular methods are used to realize variable separation for nonlinear evolution equations. We take the (2 + 1)-dimensional modified Broer–Kaup system for an example to illustrate this point. All solutions obtained by the extended projective Ricatti equation method and the improved tanh-function method coincide with the one obtained by the multilinear variable separation approach. Moreover, based on one of variable separation solutions, we also find that although abundant localized coherent structures can be constructed for a special component, we must pay our attention to the solution expression of the corresponding other component for the same equation lest many un-physical related structures might be obtained.

[1]  Ding-jiang Huang,et al.  Variable-coefficient projective Riccati equation method and its application to a new (2 + 1)-dimensional simplified generalized Broer–Kaup system , 2005 .

[2]  N. Kudryashov,et al.  Popular ansatz methods and solitary wave solutions of the Kuramoto-Sivashinsky equation , 2009 .

[3]  Anjan Biswas,et al.  Bright and dark optical solitons with time-dependent coefficients in a non-Kerr law media , 2010 .

[4]  Shun-dong Zhu,et al.  The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional Boiti–Leon–Pempinelle equation , 2008 .

[5]  Zhang Jie-fang,et al.  Variable Separation Solutions in (1+1)-Dimensional and (3+1)-Dimensional Systems via Entangled Mapping Approach , 2006 .

[6]  Zhenhai Liu,et al.  New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity via modified trigonometric function series method , 2011 .

[7]  C. Dai,et al.  Variable separation approach by means of the mapping method , 2007 .

[8]  C. Zheng,et al.  New Variable Separation Excitations of a (2+1)-Dimensional Broer-Kaup-Kupershmidt System Obtained by an Extended Mapping Approach , 2004 .

[9]  Zheng Chun-long,et al.  Embedded-Soliton and Complex Wave Excitations of (3+1)-Dimensional Burgers System , 2008 .

[10]  Some discussions about the variable separating method for solving nonlinear models , 2010 .

[11]  C. Dai,et al.  Non-completely elastic interactions in a (2+1)-dimensional dispersive long wave equation , 2012 .

[12]  Comment on: “Application of the G′G method for the complex KdV equation” [Huiqun Zhang, Commun Nonlinear Sci Numer Simul 15;2010:1700–1704] , 2011 .

[13]  Xiao-yan Tang,et al.  Reflection and reconnection interactions of resonant dromions , 2007 .

[14]  C. Dai,et al.  New Exact Solutions of the (2+1)-Dimensional Asymmetric Nizhnik-Novikov-Veselov System , 2008 .

[15]  S. Lou,et al.  Special solutions from the variable separation approach: the Davey - Stewartson equation , 1996 .

[16]  Ying Zhang,et al.  Localized excitations in (2+1)-dimensional systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Ma Song-hua,et al.  Fractal Structures and Chaotic Behaviors in a (2 + 1)-Dimensional Nonlinear System , 2010 .

[18]  Lü Zhuo-Sheng,et al.  New Soliton-like Solutions and Multi-soliton Structures for Broer–Kaup System with Variable Coefficients* , 2005 .

[19]  Xu Chang-Zhi,et al.  Study on (2+1)-Dimensional Nizhnik?Novikov?Veselov Equation by Using Extended Mapping Approach , 2006 .

[20]  Chao-Qing Dai,et al.  The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation , 2012 .

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

[22]  Anjan Biswas,et al.  Optical solitons with log-law nonlinearity , 2010 .

[23]  Nikolai A. Kudryashov,et al.  Seven common errors in finding exact solutions of nonlinear differential equations , 2009, 1011.4268.

[24]  A Novel Class of Nonpropagating Solitary Waves in (2+1) Dimensions , 2006 .

[25]  Lina Song,et al.  A new extended Riccati equation rational expansion method and its application , 2007 .

[26]  C. Zheng.  Comments on “The generalizing Riccati equation mapping method in nonlinear evolution equation: Application to (2 + 1)-dimensional Boiti–Leon–Pempinelle equation” , 2009 .

[27]  M. A. Abdou,et al.  New periodic wave solutions via extended mapping method , 2009 .

[28]  Ahmet Bekir,et al.  The G ′ G-expansion method for the nonlinear lattice equations , 2012 .

[29]  C. Dai,et al.  NEW TYPES OF INTERACTIONS BASED ON VARIABLE SEPARATION SOLUTIONS VIA THE GENERAL PROJECTIVE RICCATI EQUATION METHOD , 2007 .

[30]  E. Abdel-salam,et al.  Quasi-periodic waves and their interactions in the (2+1)-dimensional modified dispersive water-wave system , 2009 .

[31]  N. Vitanov.  On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation , 2011 .

[32]  C. Dai,et al.  Novel variable separation solutions and exotic localized excitations via the ETM in nonlinear soliton systems , 2006 .