Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation

2001 ◽  
Vol 282 (1-2) ◽  
pp. 18-22 ◽  
Author(s):  
Engui Fan
Keyword(s):  
Kdv Equation ◽  
Soliton Solutions ◽  
Mkdv Equation ◽  
Coupled Kdv Equation

A REDUCTION mKdV METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW DOUBLY-PERIODIC SOLUTIONS FOR NONLINEAR WAVE EQUATIONS

2003 ◽  
Vol 14 (05) ◽  
pp. 661-672 ◽  
Author(s):  
ZHENYA YAN
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Wave ◽  
Symbolic Computation ◽  
Wave Equations ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Mkdv Equation ◽  
Nonlinear Wave Equations ◽  
Cubic Nonlinear Schrödinger Equation

Firstly twenty-four types of doubly-periodic solutions of the reduction mKdV equation are given. Secondly based on the reduction mKdV equation and its solutions, a systemic transformation method (called the reduction mKdV method) is developed to construct new doubly-periodic solutions of nonlinear equations. Thirdly with the aid of symbolic computation, we choose the KdV equation, the coupled variant Boussinesq equation and the cubic nonlinear Schrödinger equation to illustrate our method. As a result many types of solutions are obtained. These show that this method is simple and powerful to obtain more exact solutions including doubly-periodic solutions, soliton solutions and singly-periodic solutions to a wide class of nonlinear wave equations. Finally we further extended the method to a general form.

Approximate analytic solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled mKdV equation

2010 ◽  
Vol 19 (8) ◽  
pp. 080204 ◽  
Author(s):  
Zhao Guo-Zhong ◽  
Yu Xi-Jun ◽  
Xu Yun ◽  
Zhu Jiang ◽  
Wu Di
Keyword(s):  
Kdv Equation ◽  
Analytic Solutions ◽  
Mkdv Equation ◽  
Coupled Kdv Equation

scholarly journals Approximate Analytic Solutions of Time-Fractional Hirota-Satsuma Coupled KdV Equation and Coupled MKdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jincun Liu ◽  
Hong Li
Keyword(s):  
Kdv Equation ◽  
Taylor Series Expansion ◽  
Analytic Solutions ◽  
Mkdv Equation ◽  
Differential Transform Method ◽  
Two Dimensional ◽  
Transform Method ◽  
Coupled Equations ◽  
Differential Transform ◽  
Coupled Kdv Equation

By introducing the fractional derivative in the sense of Caputo and combining the pretreatment technique to deal with long nonlinear items, the generalized two-dimensional differential transform method is proposed for solving the time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation. The presented method is a numerical method based on the generalized Taylor series expansion which constructs an analytical solution in the form of a polynomial. The numerical results show that the generalized two-dimensional differential transform method is very effective for the fractional coupled equations.

scholarly journals The construction of the mKdV cyclic symmetric N-soliton solution by the Bäcklund transformation

2019 ◽  
Vol 34 (18) ◽  
pp. 1950136 ◽  
Author(s):  
Masahito Hayashi ◽  
Kazuyasu Shigemoto ◽  
Takuya Tsukioka
Keyword(s):  
Soliton Solution ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Mkdv Equation ◽  
Group Symmetry ◽  
Backlund Transformation ◽  
Algebraic Construction ◽  
Möbius Group ◽  
Cyclic Symmetric

We study group theoretical structures of the mKdV equation. The Schwarzian-type mKdV equation has the global Möbius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the special local Möbius transformation on the mKdV one-soliton solution which can be regarded as the commutative KdV Bäcklund transformation and can generate the mKdV cyclic symmetric N-soliton solution. In this algebraic construction to obtain multi-soliton solutions, we could observe the addition formula.

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