scholarly journals The Lattice Sine-Gordon Equation as a Superposition Formula for an NLS-Type System

2021 ◽  
Author(s):  
Dmitry K. Demskoi ◽  
Keyword(s):  
Gordon Equation ◽  
Bäcklund Transformation ◽  
Type System ◽  
Backlund Transformation ◽  
Sine Gordon Equation ◽  
Compatible System

We treat the lattice sine-Gordon equation and two of its generalised symmetries as a compatible system. Elimination of shifts from the two symmetries of the lattice sine-Gordon equation yields an integrable NLS-type system. An auto-Bäcklund transformation and a superposition formula for the NLS-type system is obtained by elimination of shifts from the lattice sine-Gordon equation and its down-shifted version. We use the obtained formulae to calculate a superposition of two and three elementary solutions.

GENERALIZED mKdV EQUATION, LIOUVILLE EQUATION, SINE-GORDON EQUATION AND SINH-GORDON EQUATION AS WELL AS A FORMAL BÄCKLUND TRANSFORMATION

2011 ◽  
Vol 25 (18) ◽  
pp. 2449-2460 ◽  
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM ◽  
JING ZHAO
Keyword(s):  
Lie Algebra ◽  
Liouville Equation ◽  
Gordon Equation ◽  
Bäcklund Transformation ◽  
Mkdv Equation ◽  
Backlund Transformation ◽  
Sine Gordon Equation ◽  
Linear Combinations ◽  
Zero Curvature ◽  
Polynomial Expansions

A Lie algebra which consists of linear combinations of one basis of the Lie algebra A1 is presented for which an isospectral Lax pair is exhibited. By using the zero curvature equation, the generalized mKdV equation, Liouville equation and sine-Gordon equation, sinh-Gordon equation are generated via polynomial expansions. Finally, we investigate a kind of formal Bäcklund transformation for the generalized sine-Gordon equation. The explicit Bäcklund transformation of the standard sine-Gordon equation is presented. The other equations given in the paper are obtained similarly.

scholarly journals Nonlocal Conservation Laws of PDEs Possessing Differential Coverings

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1760
Author(s):  
Iosif Krasil’shchik
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
Gordon Equation ◽  
Bäcklund Transformation ◽  
General Nature ◽  
Backlund Transformation ◽  
Sine Gordon Equation ◽  
Modern Language ◽  
Nonlocal Conservation Laws

In his 1892 paper, L. Bianchi noticed, among other things, that quite simple transformations of the formulas that describe the Bäcklund transformation of the sine-Gordon equation lead to what is called a nonlocal conservation law in modern language. Using the techniques of differential coverings, we show that this observation is of a quite general nature. We describe the procedures to construct such conservation laws and present a number of illustrative examples.

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