The Vector Ito–Drienfel'd–Sokolov System: Bilinear Bäcklund Transformation and Lax Pair

2011 ◽  
Vol 80 (4) ◽  
pp. 045002 ◽  
Author(s):  
Maxim Ju. Balakhnev
Keyword(s):  
Bäcklund Transformation ◽  
Lax Pair ◽  
Backlund Transformation ◽  
Bilinear Bäcklund Transformation

N-SOLITON-LIKE SOLUTIONS AND BÄCKLUND TRANSFORMATIONS FOR A NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT MODIFIED KORTEWEG-DE VRIES EQUATION

2011 ◽  
Vol 25 (05) ◽  
pp. 723-733 ◽  
Author(s):  
QIAN FENG ◽  
YI-TIAN GAO ◽  
XIANG-HUA MENG ◽  
XIN YU ◽  
ZHI-YUAN SUN ◽  
...  
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Wronskian Technique ◽  
Backlund Transformation ◽  
Bilinear Bäcklund Transformation ◽  
De Vries ◽  
Bilinear Equations

A non-isospectral and variable-coefficient modified Korteweg–de Vries (mKdV) equation is investigated in this paper. Starting from the Ablowitz–Kaup–Newell–Segur procedure, the Lax pair is established and the Bäcklund transformation in original variables is also derived. By a dependent variable transformation, the non-isospectral and variable-coefficient mKdV equation is transformed into bilinear equations, by virtue of which the N-soliton-like solution is obtained. In addition, the bilinear Bäcklund transformation gives a one-soliton-like solution from a vacuum one. Furthermore, the N-soliton-like solution in the Wronskian form is constructed and verified via the Wronskian technique.

Bilinear Bäcklund transformation, Lax pair and multi-soliton solution for a vector Ramani equation

2017 ◽  
Vol 31 (12) ◽  
pp. 1750133 ◽  
Author(s):  
Junchao Chen ◽  
Bao-Feng Feng ◽  
Yong Chen
Keyword(s):  
Soliton Solution ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Backlund Transformation ◽  
Bilinear Bäcklund Transformation ◽  
Bilinear Approach

In this paper, a vector Ramani equation is proposed by using the bilinear approach. With the help of the bilinear exchange formulae, bilinear Bäcklund transformation and the corresponding Lax pair for the vector Ramani equation are derived. Besides, multi-soliton solution expressed by pfaffian is given and proved by pfaffian techniques.

The integrability of the coupled Ramani equation with binary Bell polynomials

2020 ◽  
Vol 34 (32) ◽  
pp. 2050371
Author(s):  
Xue-Dong Chai ◽  
Chun-Xia Li
Keyword(s):  
Conservation Laws ◽  
Scale Invariance ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Bell Polynomial ◽  
Bilinear Bäcklund Transformation ◽  
Binary Bell Polynomial ◽  
Binary Bell Polynomials

Binary Bell polynomial approach is applied to study the coupled Ramani equation, which is the generalization of the Ramani equation. Based on the concept of scale invariance, the coupled Ramani equation is written in terms of binary Bell polynomials of two dimensionless field variables, which leads to the bilinear coupled Ramani equation directly. As a consequence, the bilinear Bäcklund transformation, Lax pair and conservation laws are systematically constructed by virtue of binary Bell polynomials.

scholarly journals A Bäcklund transformation and nonlinear superposition formula for the Lotka-Volterra hierarchy

2002 ◽  
Vol 44 (1) ◽  
pp. 121-128 ◽  
Author(s):  
Xing-Biao Hu ◽  
Johan Springael
Keyword(s):  
Bäcklund Transformation ◽  
Lax Pair ◽  
Volterra Equations ◽  
Backlund Transformation ◽  
Nonlinear Superposition ◽  
Bilinear Bäcklund Transformation ◽  
Nonlinear Superposition Formula

AbstractA hierarchy of bilinear Lotka-Volterra equations with a unified structure is proposed. The bilinear Bäcklund transformation for this hierarchy and the corresponding canonical Lax pair are obtained. Furthermore, the nonlinear superposition formula is proved rigorously.

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