Infinite generation of soliton-like solutions for complex nonlinear evolution differential equations via the NLSE-based constructive method

Abstract Inspired by the mapping method and the direct method of symmetry reduction, we present a nonlinear Schrodinger equation-based constructive method for solving complex nonlinear evolution equations. This method can easily generate infinite soliton-like solutions of the complex nonlinear evolution equations based on the abundant solutions of the nonlinear Schrodinger equation. These solutions include multi-soliton solutions with or without background (continuous or cnoidal wave background), rational solutions and periodic solutions. As an illustration, we apply this method to solve (3 + 1)-dimensional variable-coefficient nonlinear Schrodinger equation and obtain multi-soliton solutions with continuous and cnoidal wave backgrounds.

[1]  Xiquan Fu,et al.  Evolution of the exact spatiotemporal periodic wave and soliton solutions of the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with distributed coefficients , 2011 .

[2]  Zuntao Fu,et al.  New special structures to the (2+1)-dimensional breaking soliton equations , 2011 .

[3]  Wenrui Xue,et al.  A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients , 2004 .

[4]  Shin Hj.  Soliton scattering from a finite cnoidal wave train in a fiber. , 2001 .

[5]  Fajun Yu,et al.  Multi-rogue waves for a higher-order nonlinear Schrödinger equation in optical fibers , 2013, Appl. Math. Comput..

[6]  Abdul-Majid Wazwaz.  Structures of multiple soliton solutions of the generalized, asymmetric and modified Nizhnik-Novikov-Veselov equations , 2012, Appl. Math. Comput..

[7]  B. Malomed,et al.  Spatiotemporal optical solitons , 2005 .

[8]  P. Bélanger,et al.  Bright solitons on a cw background , 1996 .

[9]  Xie Fu-ding,et al.  Computer Algebra and Solutions to the Karamoto-Sivashinsky Equation , 2005 .

[10]  Mingshu Peng,et al.  Painlevé-integrability and explicit solutions of the general two-coupled nonlinear Schrödinger system in the optical fiber communications , 2013 .

[11]  W. M. Liu,et al.  Exact solutions of the Gross-Pitaevskii equation for stable vortex modes in two-dimensional Bose-Einstein condensates , 2010 .

[12]  B. Tian,et al.  Elastic and inelastic interactions between optical spatial solitons in nonlinear optics , 2013 .

[13]  Jie-Fang Zhang,et al.  Novel interacting phenomena in (2+1) dimensional AKNS system , 2012, Appl. Math. Comput..

[14]  M. Kruskal,et al.  New similarity reductions of the Boussinesq equation , 1989 .

[15]  M. Belić,et al.  Two-dimensional accessible solitons in PT-symmetric potentials , 2012 .

[16]  Zhuosheng Lü.  A Burgers equation-based constructive method for solving nonlinear evolution equations , 2006 .

[17]  Xiaomin Wang,et al.  Application of the extended simplest equation method to the coupled Schrödinger-Boussinesq equation , 2013, Appl. Math. Comput..

[18]  C. Dai,et al.  Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation. , 2010, Optics letters.

[19]  Gui Mu,et al.  G'/G-expansion method for the fifth-order forms of KdV-Sawada-Kotera equation , 2013, Appl. Math. Comput..

[20]  Chao-Qing Dai,et al.  Controllable optical rogue waves in the femtosecond regime. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Hongqing Zhang,et al.  Analytic solutions, Darboux transformation operators and supersymmetry for a generalized one-dimensional time-dependent Schrödinger equation , 2012, Appl. Math. Comput..

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

[23]  Chirped Wave Solutions of a Generalized (3+1)-Dimensional Nonlinear Schr¨odinger Equation , 2011 .