Darboux transformation and explicit solutions for two integrable equations

A new N-fold Darboux transformation for two integrable equations is constructed with the help of a gauge transformation for the spectral problem proposed by Qiao [Z.J. Qiao, Phys. Lett. A 192 (1994) 316–322]. By the Darboux transformation, explicit soliton and multi-soliton solutions for the two equations are obtained. In particular, soliton and complexiton solutions are shown through some figures.

[1]  Yishen Li,et al.  Bidirectional soliton solutions of the classical Boussinesq system and AKNS system , 2003 .

[2]  Xuemei Li,et al.  Quasi-periodic solution of the (2+1)-dimensional Boussinesq–Burgers soliton equation , 2003 .

[3]  Bo Tian,et al.  Darboux transformation and new solutions for the Whitham–Broer–Kaup equations , 2008 .

[4]  Z. Qiao.  Generalized r-Matrix Structure and Algebro-Geometric Solution for Integrable System , 2001, nlin/0210039.

[5]  P. Clarkson,et al.  Painleve analysis of the non-linear Schrodinger family of equations , 1987 .

[6]  Zhaqilao,et al.  DARBOUX TRANSFORMATION AND VARIOUS SOLUTIONS FOR A NONLINEAR EVOLUTION EQUATION IN (3 + 1)-DIMENSIONS , 2008 .

[7]  R. Meinel,et al.  General N-soliton solution of the AKNS class on arbitrary background , 1984 .

[8]  広田 良吾,et al.  The direct method in soliton theory , 2004 .

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

[10]  Completely integrable system related to a new hierarchy of isospectral evolution equations , 1994 .

[11]  Ai-Hua Chen,et al.  Darboux transformation and multi-soliton solutions of Boussinesq–Burgers equation , 2005 .

[12]  Hong-qing Zhang,et al.  Explicit N-fold Darboux transformation and multi-soliton solutions for the (1 + 1)-dimensional higher-order Broer–Kaup system , 2007 .

[13]  Wolfgang K. Schief,et al.  Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory , 2002 .

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

[15]  Xiao-yan Tang,et al.  Variable separation solutions obtained from Darboux Transformations for the asymmetric Nizhnik–Novikov–Veselov system , 2004 .

[16]  Zixiang Zhou,et al.  Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry , 2005 .

[17]  Xianguo Geng,et al.  Darboux Transformation and Soliton Solutions for Generalized Nonlinear Schrödinger Equations , 1999 .

[18]  D. Levi,et al.  A NEW NONLINEAR SCHRODINGER EQUATION, ITS HIERARCHY AND N SOLITON SOLUTIONS , 1984 .

[19]  Zhaqilao,et al.  A generalized coupled Korteweg–de Vries hierarchy, bi-Hamiltonian structure, and Darboux transformation , 2010 .

[20]  Zixiang Zhou,et al.  Darboux Transformations in Integrable Systems , 2005 .

[21]  Z. Qiao.  Communications in Mathematical Physics The Camassa-Holm Hierarchy , N-Dimensional Integrable Systems , and Algebro-Geometric Solution on a Symplectic Submanifold , 2003 .

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

[23]  Z. Qiao,et al.  On peaked and smooth solitons for the Camassa-Holm equation , 2006 .