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.

Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid

2016 ◽  
Vol 27 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Zi-Jian Xiao ◽  
Bo Tian ◽  
Hui-Ling Zhen ◽  
Jun Chai ◽  
Xiao-Yu Wu
Keyword(s):  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Backlund Transformation

BÄCKLUND TRANSFORMATION AND ANALYTIC SOLUTIONS FOR A GENERALIZED VARIABLE-COEFFICIENT MODIFIED KORTEWEG–DE VRIES MODEL FROM FLUID DYNAMICS AND PLASMAS

2008 ◽  
Vol 19 (11) ◽  
pp. 1659-1671 ◽  
Author(s):  
FU-WEI SUN ◽  
YI-TIAN GAO ◽  
CHUN-YI ZHANG ◽  
XIAO-GE XU
Keyword(s):  
Fluid Dynamics ◽  
External Force ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Analytic Solutions ◽  
Mkdv Equation ◽  
Backlund Transformation ◽  
De Vries

We investigate a generalized variable-coefficient modified Korteweg–de Vries model with perturbed factor and external force (vc-GmKdV) describing fluid dynamics and space plasmas. In this paper, we propose an extended variable-coefficient balancing-act method (Evc-BAM), which is concise and straightforward, to obtain the generalized analytic solutions including solitary wave solution of the vc-GmKdV model with symbolic computation. Meanwhile, using the Evc-BAM, we obtain an auto-Bäcklund transformation for the vc-GmKdV model on the relevant constraint conditions of the coefficient functions. Using the given auto-Bäcklund transformation, the solutions of special equations for the vc-GmKdV model are also obtained as the variable-coefficient Korteweg–de Vries (vc-KdV) equation, the generalized KdV equation with perturbed factor and external force (GKdV), the variable-coefficient modified Korteweg–de Vries (vc-mKdV) equation, and the variable-coefficient cylindrical modified Korteweg–de Vries (vc-cmKdV) equation, respectively.

scholarly journals The Painlevé Tests, Bäcklund Transformation and Bilinear Form for the KdV Equation with a Self-Consistent Source

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yali Shen ◽  
Fengqin Zhang ◽  
Xiaomei Feng
Keyword(s):  
Soliton Solution ◽  
Bilinear Form ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Bilinear Operator ◽  
Backlund Transformation ◽  
Painlevé Property ◽  
Bilinear Bäcklund Transformation ◽  
The Kdv Equation ◽  
Self Consistent

The Painlevé property and Bäcklund transformation for the KdV equation with a self-consistent source are presented. By testing the equation, it is shown that the equation has the Painlevé property. In order to further prove its integrality, we give its bilinear form and construct its bilinear Bäcklund transformation by the Hirota's bilinear operator. And then the soliton solution of the equation is obtained, based on the proposed bilinear form.

Painlevé analysis, auto-Bäcklund transformation and analytic solutions for modified KdV equation with variable coefficients describing dust acoustic solitary structures in magnetized dusty plasmas

2021 ◽  
pp. 2150464
Author(s):  
Shailendra Singh ◽  
S. Saha Ray
Keyword(s):  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Dusty Plasmas ◽  
Mkdv Equation ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Painlevé Analysis ◽  
Backlund Transformation ◽  
Painleve Analysis

In this paper, variable coefficients mKdV equation is examined by using Painlevé analysis and auto-Bäcklund transformation method. The proposed equation is an important equation in magnetized dusty plasmas. The Painlevé analysis is used to determine the integrability whereas an auto-Bäcklund transformation technique is being explored to derive unique family of analytical solutions for variable coefficients mKdV equation. New kink–antikink and periodic-kink- type soliton solutions have been determined successfully for the considered equation. This paper shows that auto-Bäcklund transformation method is effective, direct and easy to use, and used to determine the analytic soliton solutions of various nonlinear evolution equations in the field of science and engineering. The results are plotted graphically to signify the potency and applicability of this proposed scheme for solving the above considered equation. The obtained results are in the form of soliton-like solutions, solitary wave solutions, exponential and trigonometric function solutions. Therefore, these solutions help us to understand the potential and physical behaviors of the proposed equation.

THE SECOND-ORDER KdV EQUATION AND ITS SOLITON-LIKE SOLUTION

2009 ◽  
Vol 23 (14) ◽  
pp. 1771-1780 ◽  
Author(s):  
CHUN-TE LEE ◽  
JINN-LIANG LIU ◽  
CHUN-CHE LEE ◽  
YAW-HONG KANG
Keyword(s):  
Solitary Waves ◽  
Soliton Solution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Second Order ◽  
Close Resemblance ◽  
De Vries

This paper presents both the theoretical and numerical explanations for the existence of a two-soliton solution for a second-order Korteweg-de Vries (KdV) equation. Our results show that there exists "quasi-soliton" solutions for the equation in which solitary waves almost retain their identities in a suitable physical regime after they interact, and bear a close resemblance to the pure KdV solitons.

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