A Special Integrable Differential-Difference Equation and Its Related Systems: Bilinear Forms Soliton Solutions and Lax Pairs

2003 ◽  
Vol 72 (2) ◽  
pp. 265-272 ◽  
Author(s):  
Hon-Wah Tam ◽  
Xing-Biao Hu
Keyword(s):  
Difference Equation ◽  
Soliton Solutions ◽  
Bilinear Forms ◽  
Lax Pairs ◽  
Differential Difference Equation

On the Full-Discrete Extended Generalised q-Difference Toda System

2017 ◽  
Vol 72 (8) ◽  
pp. 703-709
Author(s):  
Chuanzhong Li ◽  
Anni Meng
Keyword(s):  
Difference Equation ◽  
Hamiltonian Structure ◽  
Soliton Solutions ◽  
Toda Equation ◽  
Lax Pairs ◽  
Toda System ◽  
Toda Hierarchy

AbstractIn this paper, we construct a full-discrete integrable difference equation which is a full-discretisation of the generalised q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of an extended generalised full-discrete q-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the extended full-discrete generalised q-Toda hierarchy are given.

scholarly journals Rational-Like Solutions of a Differential-Difference Equation Related to Ablowitz-Ladik Spectral Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Jiang-ping Zhang ◽  
Qi Li ◽  
Shou-ting Chen
Keyword(s):  
Difference Equation ◽  
Spectral Problem ◽  
Matrix Equation ◽  
Soliton Solutions ◽  
General Solutions ◽  
Special Cases ◽  
Differential Difference Equation

By using the Casoratian technique, we construct the double Casoratian solutions whose entries satisfy matrix equation of a differential-difference equation related to the Ablowitz-Ladik spectral problem. Soliton solutions and rational-like solutions are obtained from taking special cases in general solutions.

Constructing Quasi-Periodic Wave Solutions of Differential-Difference Equation by Hirota Bilinear Method

2016 ◽  
Vol 71 (12) ◽  
pp. 1159-1165
Author(s):  
Qi Wang
Keyword(s):  
Difference Equation ◽  
Wave Solution ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Differential Difference Equation ◽  
Hirota Bilinear

AbstractIn the present paper, based on the Riemann theta function, the Hirota bilinear method is extended to directly construct a kind of quasi-periodic wave solution of a new integrable differential-difference equation. The asymptotic property of the quasi-periodic wave solution is analyzed in detail. It will be shown that quasi-periodic wave solution converge to the soliton solutions under certain conditions and small amplitude limit.

Construction of Quasi-Periodic Wave Solutions for Differential- Difference Equation

2012 ◽  
Vol 67 (1-2) ◽  
pp. 21-28 ◽  
Author(s):  
Y. C. Hon ◽  
Qi Wang
Keyword(s):  
Difference Equation ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Constructive Method ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Equation Analysis ◽  
Differential Difference Equation ◽  
Generalized Differential

Based on the use of the Hirota bilinear method and the Riemann theta function, we develop in this paper a constructive method for obtaining explicit quasi-periodic wave solutions of a new integrable generalized differential-difference equation. Analysis on the asymptotic property of the quasiperiodic wave solutions is given, and it is shown that the quasi-periodic wave solutions converge to the soliton solutions under certain conditions.

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