Characters of Explicit Solutions for a Semidiscrete Integrable Coupled Equation

Qiulan Zhao,Xiangwen Qu,Qianqian Yang,Wen-Xiu Ma

doi:10.1155/2021/9964540

Abstract

A semidiscrete integrable coupled system is obtained by embedding a free function into the discrete zero-curvature equation. Then, explicit solutions of the first two nontrivial equations in this system are derived directly by the Darboux transformation method. Finally, in order to compare the solutions before and after coupling intuitively, their structure figures are presented and analyzed.

Highlights

Integrable coupled equations have attracted more attention in soliton theory in recent years
For a given integrable system, we can construct a nontrivial system of differential equations which is still integrable and includes the original integrable system as a subsystem [1]
Fuchssteiner in proposed the important question: how should completely systems interact without losing complete integrability [3]? the method for constructing integrable coupling systems by perturbation was first proposed by Ma and Fuchssteiner [4]

Summary

Introduction

Integrable coupled equations have attracted more attention in soliton theory in recent years. It mainly includes perturbations, enlarging spectral problems [5, 6], creating new loop algebras [7], and multi-integrable couplings. Darboux transformation is a useful tool for solving integrable equations It can obtain its nontrivial solutions in accordance with an arbitrary seed solution of the integrable equations. Solving integrable coupled equations by the Darboux transformation method is a meaningful investigation. Explicit solutions of an integrable coupled system of Merola-Ragnisco-Tu lattice equation (24) are investigated, and explicit solutions of a new discrete integrable soliton hierarchy with 4 × 4 Lax pair [20] are discussed by the Darboux transformation. We will concentrate on investigating explicit solutions of Equation (1) by means of the Darboux transformation method based on its Lax pair.

A New Integrable Coupled Equation and Its

B T T T T C

Explicit Solutions

Conclusions

Conflicts of Interest