scholarly journals A Family of Integrable Different-Difference Equations, Its Hamiltonian Structure, and Darboux-Bäcklund Transformation

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Xi-Xiang Xu ◽  
Meng Xu
Keyword(s):  
Exact Solution ◽  
Difference Equations ◽  
Hamiltonian Structure ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Liouville Integrability ◽  
Backlund Transformation ◽  
Zero Curvature Representation ◽  
Zero Curvature ◽  
Matrix Spectral Problem

An integrable family of the different-difference equations is derived from a discrete matrix spectral problem by the discrete zero curvature representation. Hamiltonian structure of obtained integrable family is established. Liouville integrability for the obtained family of discrete Hamiltonian systems is proved. Based on the gauge transformation between the Lax pair, a Darboux-Bäcklund transformation of the first nonlinear different-difference equation in obtained family is deduced. Using this Darboux-Bäcklund transformation, an exact solution is presented.

scholarly journals Positive and Negative Integrable Hierarchies: Bi-Hamiltonian Structure and Darboux–Bäcklund Transformation

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Hamiltonian Structure ◽  
Integrable Hierarchies ◽  
Bäcklund Transformation ◽  
Liouville Integrability ◽  
Backlund Transformation ◽  
Hamiltonian Structures ◽  
Negative Power ◽  
Zero Curvature ◽  
Matrix Spectral Problem ◽  
Hamiltonian Operators

Two integrable hierarchies are derived from a novel discrete matrix spectral problem by discrete zero curvature equations. They correspond, respectively, to positive power and negative power expansions of Lax operators with respect to the spectral parameter. The bi-Hamiltonian structures of obtained hierarchies are established by a pair of Hamiltonian operators through discrete trace identity. The Liouville integrability of the obtained hierarchies is proved. Through a gauge transformation of the Lax pair, a Darboux–Bäcklund transformation is constructed for the first nonlinear different-difference equation in the negative hierarchy. Ultimately, applying the obtained Darboux–Bäcklund transformation, two exact solutions are given by means of mathematical software.

scholarly journals A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Spectral Problem ◽  
Lie Algebra ◽  
Difference Equations ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Zero Curvature Representation ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Liouville Integrable

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

scholarly journals A Soliton Hierarchy Associated with a Spectral Problem of 2nd Degree in a Spectral Parameter and Its Bi-Hamiltonian Structure

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Yuqin Yao ◽  
Shoufeng Shen ◽  
Wen-Xiu Ma
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Liouville Integrability ◽  
Hamiltonian Structures ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Matrix Spectral Problem

Associated withso~(3,R), a new matrix spectral problem of 2nd degree in a spectral parameter is proposed and its corresponding soliton hierarchy is generated within the zero curvature formulation. Bi-Hamiltonian structures of the presented soliton hierarchy are furnished by using the trace identity, and thus, all presented equations possess infinitely commuting many symmetries and conservation laws, which implies their Liouville integrability.

scholarly journals Integrability in Nonlinear Realization Scheme

1997 ◽  
Vol 12 (01) ◽  
pp. 231-236 ◽  
Author(s):  
R. P. Malik
Keyword(s):  
Hamiltonian Structure ◽  
Geometric Approach ◽  
Boussinesq Equations ◽  
Lax Pair ◽  
Conserved Quantities ◽  
Nonlinear Realization ◽  
Zero Curvature Representation ◽  
Zero Curvature ◽  
Key Features

In the framework of universal geometric approach of nonlinear realization method, some of the key features of the integrability properties of the Boussinesq equations, connected with the W3 algebra of Zamolodchikov, are discussed. The geometrical origins for these equations, its second Hamiltonian structure, Lax-pair formulation, zero-curvature representation, involving conserved quantities, etc., have also been concisely dealt with under the nonlinear realization scheme.

Complete Integrability Prolongation Structure and Backlund Transformation for the Konno-Onno Equation

2000 ◽  
Vol 55 (5) ◽  
pp. 545-549
Author(s):  
Chandan Kr. Das ◽  
A. Roy Chowdhury
Keyword(s):  
Positive Result ◽  
Explicit Form ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Independent Study ◽  
Complete Integrability ◽  
Painlevé Test ◽  
Backlund Transformation ◽  
Prolongation Structure ◽  
Prolongation Theory

Abstract Painleve analysis is used to study the complete integrability of the recently proposed Konno-Onno equation, which also leads to a general form of solutions of the system. An independent study, using the prolongation theory, gives the explicit form of the Lax pair which is then used to obtain the Backlund transformation connecting two sets of solutions of the system. The existence of the Lax pair and the positive result of the Painleve test indicate the complete integrability of the system

EXACT SOLUTIONS FOR DIFFERENTIAL-DIFFERENCE EQUATIONS BY BÄCKLUND TRANSFORMATION OF RICCATI EQUATION

2010 ◽  
Vol 24 (27) ◽  
pp. 2713-2724
Author(s):  
Y. C. HON ◽  
YUFENG ZHANG ◽  
JIANQIN MEI
Keyword(s):  
Riccati Equation ◽  
Difference Equations ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Hyperbolic Function ◽  
Backlund Transformation ◽  
Lattice Equation ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions

Based on a Bäcklund transformation of the Riccati equation and its known soliton solutions, we obtain in this paper some exact traveling-wave solutions, including triangle function solutions and hyperbolic function solutions, of a hybrid lattice equation. The proposed method can be easily extended to locate exact solitary wave solutions for other types of differential-difference equations.

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