Painlevé property, Bäcklund transformation, Lax pair and soliton-like solution for a variable coefficient KP equation

1993 ◽  
Vol 182 (2-3) ◽  
pp. 277-281 ◽  
Author(s):  
Zuo-nong Zhu
Keyword(s):  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Backlund Transformation ◽  
Kp Equation ◽  
Painlevé Property

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.

Solitons, Bäcklund transformation and Lax pair for a generalized variable-coefficient Boussinesq system in the two-layered fluid flow

2016 ◽  
Vol 30 (32n33) ◽  
pp. 1650383 ◽  
Author(s):  
Xue-Hui Zhao ◽  
Bo Tian ◽  
Jun Chai ◽  
Yu-Xiao Wu ◽  
Yong-Jiang Guo
Keyword(s):  
Fluid Flow ◽  
Water Waves ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Variable Coefficients ◽  
Boussinesq System ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Layered Fluid

Under investigation in this paper is a generalized variable-coefficient Boussinesq system, which describes the propagation of the shallow water waves in the two-layered fluid flow. Bilinear forms, Bäcklund transformation and Lax pair are derived by virtue of the Bell polynomials. Hirota method is applied to construct the one- and two-soliton solutions. Propagation and interaction of the solitons are illustrated graphically: kink- and bell-shape solitons are obtained; shapes of the solitons are affected by the variable coefficients [Formula: see text], [Formula: see text] and [Formula: see text] during the propagation, kink- and anti-bell-shape solitons are obtained when [Formula: see text], anti-kink- and bell-shape solitons are obtained when [Formula: see text]; Head-on interaction between the two bidirectional solitons, overtaking interaction between the two unidirectional solitons are presented; interactions between the two solitons are elastic.

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