scholarly journals Darboux transformation and explicit solutions for two integrable equations

2011 ◽  
Vol 380 (2) ◽  
pp. 794-806 ◽  
Author(s):  
Zhaqilao ◽  
Zhijun Qiao
Keyword(s):  
Darboux Transformation ◽  
Explicit Solutions ◽  
Integrable Equations

scholarly journals ON THE APPLICATION OF A GENERALIZED VERSION OF THE DRESSING METHOD TO THE INTEGRATION OF VARIABLE COEFFICIENT N-COUPLED NONLINEAR SCHRÖDINGER EQUATION

2012 ◽  
Vol 19 (04) ◽  
pp. 1250028
Author(s):  
TING SU ◽  
HUIHUI DAI ◽  
XIAN GUO GENG
Keyword(s):  
Optical Fibers ◽  
Variable Coefficient ◽  
High Order ◽  
Explicit Solutions ◽  
Compatibility Conditions ◽  
Dressing Method ◽  
Integrable Equations ◽  
Nonlinear Schrödinger ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Hirota Equations

N-coupled nonlinear Schrödinger (NLS) equations have been proposed to describe N-pulse simultaneous propagation in optical fibers. When the fiber is nonuniform, N-coupled variable-coefficient NLS equations can arise. In this paper, a family of N-coupled integrable variable-coefficient NLS equations are studied by using a generalized version of the dressing method. We first extend the dressing method to the versions with (N + 1) × (N + 1) operators and (2N + 1) × (2N + 1) operators. Then, we obtain three types of N-coupled variable-coefficient equations (N-coupled NLS equations, N-coupled Hirota equations and N-coupled high-order NLS equations). Then, the compatibility conditions are given, which insure that these equations are integrable. Finally, the explicit solutions of the new integrable equations are obtained.

Explicit Solutions and Conservation Laws of a Coupled Burgers’ Equation

2017 ◽  
Vol 72 (9) ◽  
pp. 789-793
Author(s):  
Bo Xue ◽  
Fang Li ◽  
Yihao Li ◽  
Mingming Sun
Keyword(s):  
Conservation Laws ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Burgers Equation ◽  
Integrable Equation ◽  
Explicit Solutions ◽  
Spectral Problems ◽  
Coupled Burgers Equation

AbstractBased on the gauge transformation between the corresponding 3×3 matrix spectral problems, N-fold Darboux transformation for a coupled Burgers’ equation is constructed. Considering the N=1 case of the derived Darboux transformation, explicit solutions for the coupled Burgers’ equation are given and their figures are plotted. Moreover, conservation laws of this integrable equation are deduced.

scholarly journals N-fold Darboux transformation and explicit solutions in terms of the determinant for the three-field Blaszak–Marciniak lattice

2013 ◽  
Vol 26 (11) ◽  
pp. 1076-1081 ◽  
Author(s):  
Xiao-Yong Wen ◽  
Xiang-Hua Meng ◽  
Xiao-Ge Xu ◽  
Jing-Tao Wang
Keyword(s):  
Darboux Transformation ◽  
Explicit Solutions

DARBOUX TRANSFORMATION AND EXPLICIT SOLUTIONS FOR A 3-FIELD INTEGRABLE LATTICE SYSTEM WITH THREE ARBITRARY CONSTANTS

2011 ◽  
Vol 25 (19) ◽  
pp. 2609-2619 ◽  
Author(s):  
XI-XIANG XU
Keyword(s):  
Darboux Transformation ◽  
Lax Pair ◽  
Lattice System ◽  
Explicit Solutions ◽  
Integrable Lattice ◽  
In Virtue Of

A 3-field integrable lattice system with three arbitrary constants and its Lax pair are presented. In virtue of the Lax pair, a Darboux transformation for the 3-field integrable lattice system is obtained, from which the explicit solutions of the 3-field integrable lattice system are given.

Quasi-Grammian solutions of the generalized Heisenberg magnet model

2020 ◽  
Vol 98 (3) ◽  
pp. 303-311
Author(s):  
Z. Amjad ◽  
B. Haider
Keyword(s):  
Darboux Transformation ◽  
Lie Group ◽  
Soliton Solutions ◽  
Two Dimensions ◽  
Explicit Solutions ◽  
Heisenberg Magnet ◽  
Model Based ◽  
Binary Darboux Transformation

In this paper we use standard binary Darboux transformation to obtain the quasi-Grammian multi-soliton solutions of generalized Heisenberg magnet model in two dimensions. We also discuss the model based on the Lie group SU(n) and obtain explicit solutions of the model for the SU(2) case.

scholarly journals Characters of Explicit Solutions for a Semidiscrete Integrable Coupled Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Qiulan Zhao ◽  
Qianqian Yang ◽  
Xiangwen Qu
Keyword(s):  
Darboux Transformation ◽  
Coupled System ◽  
Transformation Method ◽  
Explicit Solutions ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Before And After ◽  
Free Function ◽  
Coupled Equation

A semidiscrete integrable coupled system is obtained by embedding a free function into the discrete zero-curvature equation. Then, explicit solutions of the first two nontrivial equations in this system are derived directly by the Darboux transformation method. Finally, in order to compare the solutions before and after coupling intuitively, their structure figures are presented and analyzed.

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