Nonlocal Trends in the Geometry of Differential Equations: Symmetries, Conservation Laws, and Bäcklund Transformations

1989 ◽  
pp. 161-209 ◽  
Author(s):  
I. S. Krasil’shchik ◽  
A. M. Vinogradov
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Bäcklund Transformations ◽  
Backlund Transformations

scholarly journals Bäcklund transformations for several cases of a type of generalized KdV equation

2004 ◽  
Vol 2004 (63) ◽  
pp. 3369-3377
Author(s):  
Paul Bracken
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Generalized Kdv Equation ◽  
Backlund Transformations ◽  
De Vries ◽  
Derivative Term

An alternate generalized Korteweg-de Vries system is studied here. A procedure for generating solutions is given. A theorem is presented, which is subsequently applied to this equation to obtain a type of Bäcklund transformation for several specific cases of the power of the derivative term appearing in the equation. In the process, several interesting, new, ordinary, differential equations are generated and studied.

A new hierarchy of Korteweg–de Vries equations

1976 ◽  
Vol 351 (1666) ◽  
pp. 407-422 ◽  
Keyword(s):  
Conservation Laws ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Bäcklund Transformations ◽  
Multiple Soliton Solutions ◽  
Oscillating Solutions ◽  
Backlund Transformations ◽  
De Vries ◽  
Rank 2

We have found new hierarchies of Korteweg–de Vries and Boussinesq equations which have multiple soliton solutions. In contrast to the stan­dard hierarchy of K. de V. equations found by Lax, these equations do not appear to fit the present inverse formalism or possess the various pro­perties associated with it such as Bäcklund transformations. The most interesting of the new K. de V. equations is ( u nx ≡ ∂ n u /∂ x n ) ( u 4 x + 30 uu 2 x + 60 u 3 ) x + u t = 0. We have proved that this equation has N -soliton solutions but we have been able to find only two soliton solutions for the rest of this hierarchy. The above equation has higher conservation laws of rank 3, 4, 6 and 7 but none of rank 2, 5 and 8 and hence it would seem that an unusual series of conservation laws exists with every third one missing. Apart from the Boussinesq equation itself, which has N -soliton solutions, ( u xx + 6 u 2 ) xx + u xx – u tt = 0 we have found only two-soliton solutions to the rest of this second class. The new equations have bounded oscillating solutions which do not occur for the K. de V. equation itself.

Solitons, Bäcklund transformations, Lax pair and conservation laws for the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients

2016 ◽  
Vol 30 (03) ◽  
pp. 1650008 ◽  
Author(s):  
Lei Liu ◽  
Bo Tian ◽  
Wen-Rong Sun ◽  
Yu-Feng Wang ◽  
Yun-Po Wang
Keyword(s):  
Conservation Laws ◽  
Gordon Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Time Dependent ◽  
Bäcklund Transformations ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Bell Polynomial ◽  
Backlund Transformations

The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg–de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Bäcklund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz–Kaup–Newell–Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well.

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