scholarly journals Lax pair, Bäcklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation

2007 ◽  
Vol 336 (2) ◽  
pp. 1443-1455 ◽  
Author(s):  
Juan Li ◽  
Tao Xu ◽  
Xiang-Hua Meng ◽  
Ya-Xing Zhang ◽  
Hai-Qiang Zhang ◽  
...  
Keyword(s):  
Plasma Physics ◽  
Symbolic Computation ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Ocean Dynamics ◽  
Backlund Transformation ◽  
Gardner Equation ◽  
Nonlinear Lattice

Truncated Painlevé Expansion with Symbolic Computation for a General Kadomtsev-Petviashvili Equation with Variable Coefficients

1996 ◽  
Vol 51 (3) ◽  
pp. 175-178
Author(s):  
Bo Tian ◽  
Yi-Tian Gao
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Variable Coefficients ◽  
Explicit Solutions ◽  
Current Interest ◽  
Backlund Transformation ◽  
Petviashvili Equation ◽  
Mathematical Sciences ◽  
Truncated Painlevé Expansion

Able to realistically model various physical situations, the variable-coefficient generalizations of the celebrated Kadmotsev-Petviashvili equation are of current interest in physical and mathematical sciences. In this paper, we make use of both the truncated Painleve expansion and symbolic computation to obtain an auto-Bäcklund transformation and certain soliton-typed explicit solutions for a general Kadomtsev-Petviashvili equation with variable coefficients.

Bäcklund transformations, Lax pair and solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves

2021 ◽  
Author(s):  
Tian-Yu Zhou ◽  
Bo Tian ◽  
Su-Su Chen ◽  
Cheng-Cheng Wei ◽  
Yu-Qi Chen
Keyword(s):  
Burgers Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Ocean Dynamics ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Dispersive Waves ◽  
Backlund Transformations ◽  
Hybrid Solutions ◽  
Kink Waves

Burgers-type equations are considered as the models of certain phenomena in plasma astrophysics, ocean dynamics, atmospheric science and so on. In this paper, a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves is studied. Based on the Painlevé-Bäcklund equations, one auto-Bäcklund transformation and two hetero-Bäcklund transformations are derived. Motivated by the Burgers hierarchy, a Lax pair is given. Via two hetero-Bäcklund transformations with different constant seed solutions, we find some multiple-kink solutions, complex periodic solutions, hybrid solutions composed of the lump, periodic and multiple kink waves. Then we discuss the influence of the coefficients of the above equation on such solutions. Via the auto-Bäcklund transformation with the nontrivial seed solutions, we obtain certain lump-type solutions, kink-type solutions and recurrence relation of the above equation.

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.

Symbolic Computation Study of a Generalized Variable-Coefficient Two-Dimensional Korteweg-de Vries Model with Various External-Force Terms from Shallow Water Waves, Plasma Physics, and Fluid Dynamics

2009 ◽  
Vol 64 (3-4) ◽  
pp. 222-228 ◽  
Author(s):  
Xing Lü ◽  
Li-Li Li ◽  
Zhen-Zhi Yao ◽  
Tao Geng ◽  
Ke-Jie Cai ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
External Force ◽  
Bilinear Form ◽  
Water Waves ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Two Dimensional ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
De Vries

Abstract The variable-coefficient two-dimensional Korteweg-de Vries (KdV) model is of considerable significance in describing many physical situations such as in canonical and cylindrical cases, and in the propagation of surface waves in large channels of varying width and depth with nonvanishing vorticity. Under investigation hereby is a generalized variable-coefficient two-dimensional KdV model with various external-force terms. With the extended bilinear method, this model is transformed into a variable-coefficient bilinear form, and then a Bäcklund transformation is constructed in bilinear form. Via symbolic computation, the associated inverse scattering scheme is simultaneously derived on the basis of the aforementioned bilinear Bäcklund transformation. Certain constraints on coefficient functions are also analyzed and finally some possible cases of the external-force terms are discussed

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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