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 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.

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.

N-SOLITON-LIKE SOLUTIONS AND BÄCKLUND TRANSFORMATIONS FOR A NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT MODIFIED KORTEWEG-DE VRIES EQUATION

2011 ◽  
Vol 25 (05) ◽  
pp. 723-733 ◽  
Author(s):  
QIAN FENG ◽  
YI-TIAN GAO ◽  
XIANG-HUA MENG ◽  
XIN YU ◽  
ZHI-YUAN SUN ◽  
...  
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Wronskian Technique ◽  
Backlund Transformation ◽  
Bilinear Bäcklund Transformation ◽  
De Vries ◽  
Bilinear Equations

A non-isospectral and variable-coefficient modified Korteweg–de Vries (mKdV) equation is investigated in this paper. Starting from the Ablowitz–Kaup–Newell–Segur procedure, the Lax pair is established and the Bäcklund transformation in original variables is also derived. By a dependent variable transformation, the non-isospectral and variable-coefficient mKdV equation is transformed into bilinear equations, by virtue of which the N-soliton-like solution is obtained. In addition, the bilinear Bäcklund transformation gives a one-soliton-like solution from a vacuum one. Furthermore, the N-soliton-like solution in the Wronskian form is constructed and verified via the Wronskian technique.

A new explicit solution to the lattice sine-Gordon equation

2016 ◽  
Vol 30 (09) ◽  
pp. 1650166 ◽  
Author(s):  
Xiaoxue Xu ◽  
Cewen Cao
Keyword(s):  
Riemann Surface ◽  
Explicit Solution ◽  
Gordon Equation ◽  
Theta Functions ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Sine Gordon Equation ◽  
Exponential Functions ◽  
Surface Method ◽  
Higher Genus

Based on a new discrete Lax pair, an elementary explicit solution is found for the lattice sine-Gordon equation through Riemann surface method. It contains only exponential functions, quite different from the usual higher genus solutions, expressed with complicated theta functions. The solutions to the associated models, the lattice potential MKdV equation and a special H3 equation are also discussed.

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