Painlevé property, Lax pair and Darboux transformation of the variable-coefficient modified Kortweg-de Vries model in fluid-filled elastic tubes

2011 ◽  
Vol 16 (4) ◽  
pp. 1776-1782 ◽  
Author(s):  
Xiao-Ling Gai ◽  
Yi-Tian Gao ◽  
Lei Wang ◽  
De-Xin Meng ◽  
Xing Lü ◽  
...  
Keyword(s):  
Darboux Transformation ◽  
Variable Coefficient ◽  
Lax Pair ◽  
Elastic Tubes ◽  
Painlevé Property ◽  
De Vries

Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation with Symbolic Computation

2008 ◽  
Vol 49 (4) ◽  
pp. 833-838 ◽  
Author(s):  
Zhang Ya-Xing ◽  
Zhang Hai-Qiang ◽  
Li Juan ◽  
Xu Tao ◽  
Zhang Chun-Yi ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
Darboux Transformation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Lax Pair ◽  
De Vries ◽  
Fifth Order

On the integrability properties of variable coefficient Korteweg – de Vries equations

1996 ◽  
Vol 74 (9-10) ◽  
pp. 676-684 ◽  
Author(s):  
F. Güngör ◽  
M. Sanielevici ◽  
P. Winternitz
Keyword(s):  
Differential Equations ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Symmetry Algebra ◽  
Time Symmetry ◽  
Kdv Equations ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

All variable coefficient Korteweg – de Vries (KdV) equations with three-dimensional Lie point symmetry groups are investigated. For such an equation to have the Painlevé property, its coefficients must satisfy seven independent partial differential equations. All of them are satisfied only for equations equivalent to the KdV equation itself. However, most of them are satisfied in all cases. If the symmetry algebra is either simple, or nilpotent, then the equations have families of single-valued solutions depending on two arbitrary functions of time. Symmetry reduction is used to obtain particular solutions. The reduced ordinary differential equations are classified.

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.

Darboux transformation and the exact solutions of the (2+1)-dimensional nonlocal complex modified Korteweg-de Vries and Maxwell–Bloch equations

2020 ◽  
Vol 34 (22) ◽  
pp. 2050230
Author(s):  
Na-Na Li ◽  
Hui-Qin Hao ◽  
Rui Guo
Keyword(s):  
Exact Solutions ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Darboux Transformations ◽  
Bloch Equations ◽  
Dynamic Behaviors ◽  
De Vries ◽  
Maxwell Bloch Equations

In this paper, we consider the (2[Formula: see text]+[Formula: see text]1)-dimensional nonlocal complex modified Korteweg-de Vries and Maxwell–Bloch (cmKdV-MB) equations. According to the relevant Lax pair presented, we construct one- and two-fold Darboux transformations (DT). The exact solutions are derived from the trivial seeds by DT and the dynamic behaviors of soliton solutions are analyzed by individual pictures.

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