The Integrable Defect System And Differential-difference Equation Related To The Darboux Transformations Of The Kaup-Newell Equation

Darboux transformations are an important research topic in integrable systems,which not only can be used to construct exact solutions of integrable systems,but also can be used to generate new integrable systems.This thesis focus on the Darboux transforms of Kaup-Newell equation.A new defect Kaup-Newell equation is proposed and a Darboux transform,as well as an exact solution,of a lattice equation of type II is obtained.The thesis is divided into three chapters.In the second chapter,we first review the construction of the Kaup-Newell equation and its conservation laws,and then give a series of Darboux-B¨acklund transformations of Kaup-Newell equation and the corresponding differential-difference equations.In the third chapter,based on the general method of constructing the defect soliton equation from the Lax representation of the soliton equation,a new defect Kaup-Newell equation,together with an infinite number of its conserved laws is constructed.In the fourth chapter,we study the Darboux transform of a lattice equation of type II.It is shown first that the lattice equation is just the second member of the relativistic Toda hierarchy of equations and that the lattice allows the Kaup-Newell spectral problem as its continuous space spectrum problem,and a Darboux transform of KaupNewell equation as its discrete time spectrum problem.A Darboux transform of the second type lattice equation is then constructed by using these spectral problems.Finally,as an application of the Darboux transform,an exact solution of this second type of lattice equation is obtained.