N-SOLITON SOLUTIONS, AUTO-BÄCKLUND TRANSFORMATIONS AND LAX PAIR FOR A NONISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION VIA SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (10) ◽  
pp. 2383-2393 ◽  
Author(s):  
LI-LI LI ◽  
BO TIAN ◽  
CHUN-YI ZHANG ◽  
HAI-QIANG ZHANG ◽  
JUAN LI ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Bäcklund Transformations ◽  
Nonlinear Superposition ◽  
Backlund Transformations ◽  
De Vries

In this paper, a nonisospectral and variable-coefficient Korteweg-de Vries equation is investigated based on the ideas of the variable-coefficient balancing-act method and Hirota method. Via symbolic computation, we obtain the analytic N-soliton solutions, variable-coefficient bilinear form, auto-Bäcklund transformations (in both the bilinear form and Lax pair form), Lax pair and nonlinear superposition formula for such an equation in explicit form. Moreover, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.

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

VARIABLE-COEFFICIENT MIURA TRANSFORMATIONS AND INTEGRABLE PROPERTIES FOR A GENERALIZED VARIABLE-COEFFICIENT KORTEWEG–de VRIES EQUATION FROM BOSE–EINSTEIN CONDENSATES WITH SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (04) ◽  
pp. 571-584 ◽  
Author(s):  
JUAN LI ◽  
BO TIAN ◽  
XIANG-HUA MENG ◽  
TAO XU ◽  
CHUN-YI ZHANG ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Kdv Equation ◽  
Nonlinear Superposition ◽  
Modified Kdv Equation ◽  
De Vries ◽  
The One ◽  
Bose Einstein ◽  
Miura Transformations

In this paper, a generalized variable-coefficient Korteweg–de Vries (KdV) equation with the dissipative and/or perturbed/external-force terms is investigated, which arises in arterial mechanics, blood vessels, Bose gases of impenetrable bosons and trapped Bose–Einstein condensates. With the computerized symbolic computation, two variable-coefficient Miura transformations are constructed from such a model to the modified KdV equation under the corresponding constraints on the coefficient functions. Meanwhile, through these two transformations, a couple of auto-Bäcklund transformations, nonlinear superposition formulas and Lax pairs are obtained with the relevant constraints. Furthermore, the one- and two-solitonic solutions of this equation are explicitly presented and the physical properties and possible applications in some fields of these solitonic structures are discussed and pointed out.

ANALYTICAL INVESTIGATION OF THE CAUDREY–DODD–GIBBON–KOTERA–SAWADA EQUATION USING SYMBOLIC COMPUTATION

2013 ◽  
Vol 27 (06) ◽  
pp. 1250124 ◽  
Author(s):  
XIAO-GE XU ◽  
XIANG-HUA MENG ◽  
CHUN-YI ZHANG ◽  
YI-TIAN GAO
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Analytical Investigation ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Nonlinear Superposition ◽  
Hirota Bilinear ◽  
Nonlinear Superposition Formula

In this paper, the Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation is analytically investigated using the Hirota bilinear method. Based on the bilinear form of the CDGKS equation, its N-soliton solution in explicit form is derived with the aid of symbolic computation. Besides the soliton solutions, several integrable properties such as the Bäcklund transformation, the Lax pair and the nonlinear superposition formula are also derived for the CDGKS equation.

ANALYTIC ANALYSIS ON A GENERALIZED (2+1)-DIMENSIONAL VARIABLE-COEFFICIENT KORTEWEG–DE VRIES EQUATION USING SYMBOLIC COMPUTATION

2010 ◽  
Vol 24 (27) ◽  
pp. 5359-5370 ◽  
Author(s):  
CHENG ZHANG ◽  
BO TIAN ◽  
LI-LI LI ◽  
TAO XU
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Nonlinear Superposition ◽  
Hirota Bilinear Form ◽  
De Vries ◽  
The One ◽  
Dimensional Variable ◽  
Hirota Bilinear ◽  
Nonlinear Superposition Formula

With the help of symbolic computation, a generalized (2+1)-dimensional variable-coefficient Korteweg–de Vries equation is studied for its Painlevé integrability. Then, Hirota bilinear form is derived, from which the one- and two-solitary-wave solutions with the corresponding graphic illustration are presented. Furthermore, a bilinear auto-Bäcklund transformation is constructed and the nonlinear superposition formula and Lax pair are also obtained. Finally, the analytic solution in the Wronskian form is constructed and proved by direct substitution into the bilinear equation.

Bäcklund transformation according to Gerdzikov and Kullish for the Korteweg–de Vries equation

1982 ◽  
Vol 60 (11) ◽  
pp. 1599-1606 ◽  
Author(s):  
Henri-François Gautrin
Keyword(s):  
Vries Equation ◽  
Bäcklund Transformation ◽  
Bäcklund Transformations ◽  
Scattering Parameters ◽  
Backlund Transformation ◽  
Specific Change ◽  
Backlund Transformations ◽  
De Vries

A study of solutions of the Gel'fand–Levitan equation permits one to establish new Bäcklund transformations for the Korteweg–de Vries equation. To a specific change in the scattering parameters, there corresponds a family of Bäcklund transformations. A means to construct these transformations is presented.

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