Bäcklund transformation, Lax pair, and solutions for the Caudrey–Dodd–Gibbon equation

2011 ◽  
Vol 52 (1) ◽  
pp. 013511 ◽  
Author(s):  
Qi-Xing Qu ◽  
Bo Tian ◽  
Kun Sun ◽  
Yan Jiang
Keyword(s):  
Bäcklund Transformation ◽  
Lax Pair ◽  
Backlund Transformation

Complete Integrability Prolongation Structure and Backlund Transformation for the Konno-Onno Equation

2000 ◽  
Vol 55 (5) ◽  
pp. 545-549
Author(s):  
Chandan Kr. Das ◽  
A. Roy Chowdhury
Keyword(s):  
Positive Result ◽  
Explicit Form ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Independent Study ◽  
Complete Integrability ◽  
Painlevé Test ◽  
Backlund Transformation ◽  
Prolongation Structure ◽  
Prolongation Theory

Abstract Painleve analysis is used to study the complete integrability of the recently proposed Konno-Onno equation, which also leads to a general form of solutions of the system. An independent study, using the prolongation theory, gives the explicit form of the Lax pair which is then used to obtain the Backlund transformation connecting two sets of solutions of the system. The existence of the Lax pair and the positive result of the Painleve test indicate the complete integrability of the system

Bäcklund transformations, Lax pair and solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves

2021 ◽  
Author(s):  
Tian-Yu Zhou ◽  
Bo Tian ◽  
Su-Su Chen ◽  
Cheng-Cheng Wei ◽  
Yu-Qi Chen
Keyword(s):  
Burgers Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Ocean Dynamics ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Dispersive Waves ◽  
Backlund Transformations ◽  
Hybrid Solutions ◽  
Kink Waves

Burgers-type equations are considered as the models of certain phenomena in plasma astrophysics, ocean dynamics, atmospheric science and so on. In this paper, a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves is studied. Based on the Painlevé-Bäcklund equations, one auto-Bäcklund transformation and two hetero-Bäcklund transformations are derived. Motivated by the Burgers hierarchy, a Lax pair is given. Via two hetero-Bäcklund transformations with different constant seed solutions, we find some multiple-kink solutions, complex periodic solutions, hybrid solutions composed of the lump, periodic and multiple kink waves. Then we discuss the influence of the coefficients of the above equation on such solutions. Via the auto-Bäcklund transformation with the nontrivial seed solutions, we obtain certain lump-type solutions, kink-type solutions and recurrence relation of the above equation.

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