A Bilinear Bäcklund Transformation and Lax Pair for a (1+1)-Dimensional Differential-Difference sine-Gordon Equation

2007 ◽  
Vol 48 (3) ◽  
pp. 487-490
Author(s):  
Qian Xian-Min ◽  
Hon-Wah Tam
Keyword(s):  
Gordon Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Backlund Transformation ◽  
Sine Gordon Equation ◽  
Bilinear Bäcklund Transformation

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.

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.

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