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

Author(s):  
Zhonglong Zhao
1982 ◽  
Vol 60 (11) ◽  
pp. 1599-1606 ◽  
Author(s):  
Henri-François Gautrin

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.


2018 ◽  
Vol 73 (3) ◽  
pp. 207-213 ◽  
Author(s):  
Rehab M. El-Shiekh

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.


1981 ◽  
Vol 81 (6) ◽  
pp. 305-309 ◽  
Author(s):  
N.C. Freeman ◽  
G. Horrocks ◽  
P. Wilkinson

2009 ◽  
Vol 23 (10) ◽  
pp. 2383-2393 ◽  
Author(s):  
LI-LI LI ◽  
BO TIAN ◽  
CHUN-YI ZHANG ◽  
HAI-QIANG ZHANG ◽  
JUAN LI ◽  
...  

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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