In the Atmosphere and Oceanic Fluids: Scaling Transformations, Bilinear Forms, Bäcklund Transformations and Solitons for A Generalized Variable-Coefficient Korteweg-de Vries-Modified Korteweg-de Vries Equation

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.

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Bäcklund transformations, nonlocal symmetry and exact solutions of a generalized (2+1)-dimensional Korteweg–de Vries equation

Chinese Journal of Physics ◽

10.1016/j.cjph.2021.07.026 ◽

2021 ◽

Author(s):

Zhonglong Zhao

Keyword(s):

Exact Solutions ◽

Vries Equation ◽

Bäcklund Transformations ◽

Nonlocal Symmetry ◽

Backlund Transformations ◽

De Vries

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

Canadian Journal of Physics ◽

10.1139/p82-215 ◽

1982 ◽

Vol 60 (11) ◽

pp. 1599-1606 ◽

Cited By ~ 1

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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Painlevé Test, Bäcklund Transformation and Consistent Riccati Expansion Solvability for two Generalised Cylindrical Korteweg-de Vries Equations with Variable Coefficients

Zeitschrift für Naturforschung A ◽

10.1515/zna-2017-0349 ◽

2018 ◽

Vol 73 (3) ◽

pp. 207-213 ◽

Cited By ~ 8

Author(s):

Rehab M. El-Shiekh

Keyword(s):

Vries Equation ◽

Dusty Plasmas ◽

Variable Coefficients ◽

Bäcklund Transformations ◽

Painlevé Test ◽

New Exact Solutions ◽

Backlund Transformations ◽

Integrability Conditions ◽

De Vries ◽

Consistent Riccati Expansion

AbstractIn this paper, the integrability of the (2+1)-dimensional cylindrical modified Korteweg-de Vries equation and the (3+1)-dimensional cylindrical Korteweg-de Vries equation with variable coefficients arising in dusty plasmas in its generalised form was studied by two different techniques: the Painlevé test and the consistent Riccati expansion solvability. The integrability conditions and Bäcklund transformations are constructed. By using Bäcklund transformations and the solutions of the Riccati equation many new exact solutions are found for the two equations in this study. Finally, the application of the obtained solutions in dusty plasmas is investigated.

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