A new Bäcklund transformation and multi-soliton solutions to the KdV equation with general variable coefficients

2001 ◽  
Vol 287 (3-4) ◽  
pp. 211-216 ◽  
Author(s):  
Mingliang Wang ◽  
Yueming Wang
Keyword(s):  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Backlund Transformation ◽  
The Kdv Equation

scholarly journals A New Auto-Bäcklund Transformation of the KdV Equation with General Variable Coefficients and Its Application

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Chunping Liu
Keyword(s):  
Solitary Wave ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Variable Coefficients ◽  
Backlund Transformation ◽  
The Kdv Equation ◽  
Homogeneity Equation ◽  
Homogeneous Balance Method ◽  
Key Steps ◽  
Homogeneous Balance

First, by improving some key steps in the homogeneous balance method, a new auto-Bäcklund transformation (BT) to the KdV equation with general variable coefficients is derived. The new auto-BT in this paper does not require the coefficients of the equation to be linearly dependent. Then, based on the new auto-BT in which there is only one quadratic homogeneity equation to be solved, an exact soliton-like solution containing 2-solitary wave is given.

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

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.

Bäcklund Transformation and Soliton Solutions for a (3+1)-Dimensional Variable-Coefficient Breaking Soliton Equation

2016 ◽  
Vol 71 (9) ◽  
pp. 797-805 ◽  
Author(s):  
Chen Zhao ◽  
Yi-Tian Gao ◽  
Zhong-Zhou Lan ◽  
Jin-Wei Yang
Keyword(s):  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Bell Polynomials ◽  
Propagation Characteristics ◽  
Backlund Transformation ◽  
Soliton Equation ◽  
Dimensional Variable ◽  
Interaction Behaviors

AbstractIn this article, a (3+1)-dimensional variable-coefficient breaking soliton equation is investigated. Based on the Bell polynomials and symbolic computation, the bilinear forms and Bäcklund transformation for the equation are derived. One-, two-, and three-soliton solutions are obtained via the Hirota method.N-soliton solutions are also constructed. Propagation characteristics and interaction behaviors of the solitons are discussed graphically: (i) solitonic direction and position depend on the sign of the wave numbers; (ii) shapes of the multisoliton interactions in the scaled space and time coordinates are affected by the variable coefficients; (iii) multisoliton interactions are elastic for that the velocity and amplitude of each soliton remain unchanged after each interaction except for a phase shift.

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.

BILINEARIZATION AND SOLUTIONS OF THE KdV6 EQUATION

2008 ◽  
Vol 06 (04) ◽  
pp. 401-412 ◽  
Author(s):  
A. RAMANI ◽  
B. GRAMMATICOS ◽  
R. WILLOX
Keyword(s):  
Evolution Equation ◽  
Arbitrary Function ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Singularity Analysis ◽  
Backlund Transformation ◽  
Integrable Evolution Equation ◽  
The Kdv Equation ◽  
Integrable Evolution ◽  
Kdv6 Equation

We examine the recently proposed KdV6 integrable evolution equation. Starting from solutions suggested by singularity analysis and using the auto-Bäcklund transformation, we construct solutions of the KdV6 which involve one arbitrary function of time. Next, we proceed to bilinearize the equation and derive a new, simpler, auto-Bäcklund transformation. Starting from the solutions of the KdV equation we construct those of the KdV6 in the form of M kinks and N poles and which indeed involve an arbitrary function of time.

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