Lax pair, auto-Bäcklund transformation and conservation law for a generalized variable-coefficient KdV equation with external-force term

2015 ◽  
Vol 45 ◽  
pp. 58-63 ◽  
Author(s):  
Yuping Zhang ◽  
Jing Liu ◽  
Guangmei Wei
Keyword(s):  
External Force ◽  
Conservation Law ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Lax Pair ◽  
Force Term ◽  
Backlund Transformation

Solitons, Bäcklund Transformation, Lax Pair, and Infinitely Many Conservation Law for a (2+1)-Dimensional Generalised Variable-Coefficient Shallow Water Wave Equation

2016 ◽  
Vol 71 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Zhong-Zhou Lan ◽  
Yi-Tian Gao ◽  
Jin-Wei Yang ◽  
Chuan-Qi Su ◽  
Da-Wei Zuo
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Conservation Law ◽  
Water Wave ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Shallow Water Wave

AbstractUnder investigation in this article is a (2+1)-dimensional generalised variable-coefficient shallow water wave equation, which describes the interaction of the Riemann wave propagating along the y axis with a long-wave propagating along the x axis in a fluid, where x and y are the scaled space coordinates. Bilinear forms, Bäcklund transformation, Lax pair, and infinitely many conservation law are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the multi-soliton interaction in the scaled space and time coordinates. (ii) Positions of the solitons depend on the sign of wave numbers after each interaction. (iii) Interaction of the solitons is elastic, i.e. the amplitude, velocity, and shape of each soliton remain invariant after each interaction except for a phase shift.

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.

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.

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