scholarly journals Infinite many conservation laws of discrete system associated with a 3×3 matrix spectral problem

2017 ◽  
Vol 21 (4) ◽  
pp. 1613-1619
Author(s):  
Sheng Zhang ◽  
Dongdong Liu
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Difference Equations ◽  
Discrete System ◽  
Nanoporous Materials ◽  
Electron Conduction ◽  
Alternative Approach ◽  
Non Linear ◽  
Linear Differential ◽  
Matrix Spectral Problem

Differential-difference equations are often considered as an alternative approach to describing some phenomena arising in heat/electron conduction and flow in carbon nanotubes and nanoporous materials. Infinite many conservation laws play important role in discussing the integrability of non-linear differential equations. In this paper, infinite many conservation laws of the non-linear differential-difference equations associated with a 3?3 matrix spectral problem are obtained.

scholarly journals Darboux transform and conservation laws of new differential-difference equations

2020 ◽  
Vol 24 (4) ◽  
pp. 2519-2527
Author(s):  
Sheng Zhang ◽  
Dongdong Liu
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Spectral Problem ◽  
Difference Equations ◽  
Non Linear ◽  
Linear Differential ◽  
Darboux Transforms

Darboux transforms, exact solutions and conservation laws are important topics in thermal science and other fields as well. In this paper, the new non-linear differential-difference equations associated a discrete linear spectral problem are studied analytically. Firstly, the Darboux transform of the studied equations is constructed, and exact solutions are then obtained. Finally, infinite many conservation laws are derived.

New hierarchy of integrable nonlinear differential-difference equations

2015 ◽  
Vol 29 (31) ◽  
pp. 1550190
Author(s):  
Xianguo Geng ◽  
Liang Guan ◽  
Bo Xue
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Difference Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Zero Curvature ◽  
Matrix Spectral Problem

A hierarchy of integrable nonlinear differential-difference equations associated with a discrete [Formula: see text] matrix spectral problem is proposed based on the discrete zero-curvature equations. Then, Hamiltonian structures for this hierarchy are constructed with the aid of the trace identity. Infinitely many conservation laws of the hierarchy are derived by means of spectral parameter expansions.

An extension of the coupled derivative nonlinear Schrödinger hierarchy

2018 ◽  
Vol 32 (02) ◽  
pp. 1850016
Author(s):  
Siqi Xu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Compatibility Condition ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Nonlinear Schrödinger ◽  
Matrix Spectral Problem

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

scholarly journals Conservation Laws and Self-Consistent Sources for an Integrable Lattice Hierarchy Associated with a Three-by-Three Discrete Matrix Spectral Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yu-Qing Li ◽  
Bao-Shu Yin
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Liouville Integrability ◽  
Hamiltonian Structures ◽  
Integrable Lattice ◽  
Self Consistent ◽  
Matrix Spectral Problem

A lattice hierarchy with self-consistent sources is deduced starting from a three-by-three discrete matrix spectral problem. The Hamiltonian structures are constructed for the resulting hierarchy. Liouville integrability of the resulting equations is demonstrated. Moreover, infinitely many conservation laws of the resulting hierarchy are obtained.

scholarly journals Analytical solutions of differential-difference sine-Gordon equation

2017 ◽  
Vol 21 (4) ◽  
pp. 1701-1705 ◽  
Author(s):  
Da-Jiang Ding ◽  
Di-Qing Jin ◽  
Chao-Qing Dai
Keyword(s):  
Elliptic Function ◽  
Variable Coefficient ◽  
Gordon Equation ◽  
Variable Coefficients ◽  
Jacobian Elliptic Function ◽  
Sine Gordon Equation ◽  
Electron Conduction ◽  
Negative Power ◽  
Non Linear ◽  
Linear Differential

In modern textile engineering, non-linear differential-difference equations are often used to describe some phenomena arising in heat/electron conduction and flow in carbon nanotubes. In this paper, we extend the variable coefficient Jacobian elliptic function method to solve non-linear differential-difference sine-Gordon equation by introducing a negative power and some variable coefficients in the ansatz, and derive two series of Jacobian elliptic function solutions. When the modulus of Jacobian elliptic function approaches to 1, some solutions can degenerate into some known solutions in the literature.

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