Bell-polynomial manipulations on the Bäcklund transformations and Lax pairs for some soliton equations with one Tau-function

2010 ◽  
Vol 51 (11) ◽  
pp. 113506 ◽  
Author(s):  
Xing Lü ◽  
Bo Tian ◽  
Kun Sun ◽  
Pan Wang
Keyword(s):  
Bäcklund Transformations ◽  
Lax Pairs ◽  
Bell Polynomial ◽  
Tau Function ◽  
Soliton Equations ◽  
Backlund Transformations

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.

BÄCKLUND TRANSFORMATIONS AND LAX PAIRS FOR THE NONLINEAR KLEIN–GORDON EQUATION WITH SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (11) ◽  
pp. 2511-2521
Author(s):  
XIAO-GE XU ◽  
YI-TIAN GAO ◽  
GUANG-MEI WEI
Keyword(s):  
Symbolic Computation ◽  
Gordon Equation ◽  
Bäcklund Transformations ◽  
Sine Gordon Equation ◽  
Lax Pairs ◽  
Special Cases ◽  
Backlund Transformations ◽  
Klein Gordon ◽  
Pulse Waves ◽  
Klein Gordon Equation

In this paper, the nonlinear Klein–Gordon equation describing the propagation of pulse waves in plasma or waveguide is investigated. With symbolic computation, the generalized Bäcklund Transformations (BTs) for this equation are constructed under different conditions. It is shown that the BTs published in the previous literature for the Sine–Gordon equation, Sinh–Gordon equation, and Liouville equation all turn out to be special cases of the results in the present paper. Moreover, the corresponding Lax pairs are explicitly derived from the obtained BTs through some transformations.

scholarly journals Bäcklund transformations of Zn-sine-Gordon systems

2017 ◽  
Vol 31 (17) ◽  
pp. 1750189 ◽  
Author(s):  
Xueping Yang ◽  
Chuanzhong Li
Keyword(s):  
Lie Algebra ◽  
General Formula ◽  
Bäcklund Transformations ◽  
Lax Pairs ◽  
Nonlinear Superposition ◽  
Commutative Subalgebra ◽  
Backlund Transformations ◽  
Algebraic Reductions ◽  
Nonlinear Superposition Formula

In this paper, from the algebraic reductions from the Lie algebra [Formula: see text] to its commutative subalgebra [Formula: see text], we construct the general [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon systems which contain many multi-component sine-Gordon type and sinh-Gordon type equations. Meanwhile, we give the Bäcklund transformations of the [Formula: see text]-sine-Gordon and [Formula: see text]-sinh-Gordon equations which can generate new solutions from seed solutions. To see the [Formula: see text]-systems clearly, we consider the [Formula: see text]-sine-Gordon and [Formula: see text]-sine-Gordon equations explicitly including their Bäcklund transformations, the nonlinear superposition formula and Lax pairs.

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