Exact Solutions, Continuous Limits Theory And Dynamical Properties Of The Solutions For Some Semi-discrete Integrable Systems

In this dissertation, by using the Darboux transformation and Hirota bilinear method, weobtain some exact solutions of some semi-discrete integrable systems and discuss dynamical prop-erties of these solutions; In order to get more insight on the relation between semi-discrete coupledintegrable systems and the continuous coupled integrable systems, we proof that the theory of somesemi-discrete integrable systems including the Lax pairs, the Darboux transformation, soliton so-lutions and conservation laws yield the corresponding theory of the continuous integrable systemsin the continuous limit. The main contents are summarized as follows:In chapter 1, we brie?y outline the research background in connection with the dissertation.In chapter 2, by considering a discrete 4×4 spectral problem, we derive a semi-discretecoupled KdV-type equation hierarchy. Our hierarchy includes the coupled Volterra system as the?rst member which is a semi-discrete version of the coupled KdV equation. We also investigatethe integrability and the multi-Hamiltonian structures for the obtained hierarchy. Furthermore, we?nd new exact solutions including multi-soliton, multi-positon, multi-negaton, and multi-periodicfor the coupled Volterra system. The dynamical properties of these new solutions are discussed indetail. We also prove that the theory of the coupled Volterra lattice system including the Lax pair,the Darboux transformation, and exact solutions yield the corresponding theory of the coupledKdV equation in the continuous limit.In chapter 3, the Darboux transformation for a Blaszak-Marciniak (BM) three-?eld and four-?eld lattice equations are constructed. As an application of the obtained Darboux transformation,explicit exact solutions of the lattice equations are given. We also discuss some properties for thesenew exact solutions. Our analysis shows that the exact solutions possess new characters.In chapter 4, we propose a new coupled semi-discrete mKdV system. The Lax pairs, the Dar-boux transformation, soliton solutions and conservation laws for the coupled semi-discrete mKdVsystem are given. The coupled mKdV theory including the Lax pairs, the Darboux transformation,soliton solutions and conservation laws is recovered through the continuous limits of correspond-ing theory for the new semi-discrete coupled mKdV system. In chapter 5, we will discuss the discrete coupled Hirota equation which is a hybrid of thediscrete coupled nonlinear Schro¨inger equation and the discrete coupled mKdV equation. Byusing Hirota direct method, exact solutions to the discrete equation are constructed. The dynamicalproperties of these solutions are discussed.