Painlevé analysis, Lax pair, Bäcklund transformation and multi-soliton solutions for a generalized variable-coefficient KdV–mKdV equation in fluids and plasmas

2012 ◽  
Vol 85 (5) ◽  
pp. 055010 ◽  
Author(s):  
Gao-Qing Meng ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Yu-Jia Shen ◽  
Yi Qin
Keyword(s):  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Painlevé Analysis ◽  
Backlund Transformation ◽  
Painleve Analysis

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.

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.

scholarly journals On Two Aspects of the Painlevé Analysis

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Sergei Sakovich
Keyword(s):  
Nonlinear Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Nonlinear Pdes ◽  
Painlevé Analysis ◽  
Painlevé Test ◽  
Backlund Transformation ◽  
Independent Variables ◽  
Painleve Analysis

We use the Calogero equation to illustrate the following two aspects of the Painlevé analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painlevé test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Bäcklund transformation related to the Lax pair.

PAINLEVÉ ANALYSIS, LAX PAIR AND BÄCKLUND TRANSFORMATION FOR THE GROSS–PITAEVSKII EQUATION IN THE BOSE–EINSTEIN CONDENSATES

2011 ◽  
Vol 25 (08) ◽  
pp. 1037-1047
Author(s):  
FENG-HUA QI ◽  
BO TIAN ◽  
TAO XU ◽  
HAI-QIANG ZHANG ◽  
LI-LI LI ◽  
...  
Keyword(s):  
Bäcklund Transformation ◽  
Scattering Length ◽  
Lax Pair ◽  
Time Dependent ◽  
Painlevé Analysis ◽  
Pitaevskii Equation ◽  
Backlund Transformation ◽  
Ansatz Method ◽  
Bose Einstein ◽  
Painleve Analysis

Due to their relevance to physics and technology, the Bose–Einstein condensates (BECs) are of current interest. Certain dynamics of the BECs, such as the cigar-shaped condensate confined in a cylindrically symmetric parabolic trap, can be described by the Gross–Pitaevskii (GP) equation with a time-dependent trap. In this paper, by virtue of the Painlevé analysis and symbolic computation, we derive the integrable condition for the GP equation with the time-dependent scattering length in the presence of a confining or expulsive time-dependent trap. Lax pair for this equation is directly obtained via the Ablowitz–Kaup–Newell–Segur scheme under the integrable condition. Bright one-soliton-like solution of the GP equation is presented via the Bäcklund transformation and some analytic solutions with variable amplitudes are obtained by the ansatz method. In addition, an infinite number of conservation laws are also derived. Those results could be of some value for the studies on the lower-dimensional condensates.

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