The Applications Of Degenerate Darboux Transformation For The Two Multi-Component Soliton Equations

In this thesis,we mainly study the two-component nonlinear Schrodinger equations and three-component Kundu-Eckhaus(KE)equations,and obtain their compact determinant representation formula of the degenerate Darboux transformation.Furthermore,we construct the expressions of the positon solutions.This thesis is divided into five chapters.In Chapter 1,we introduce the history of soliton,the research background and the main works of this thesis.In Chapter 2,we introduce the Darboux transformation and derive the degenerate Darboux transformation of the nonlinear Schrodinger equation.In Chapter 3,we formulate the formula of the degenerate Darboux transformation for the two-component nonlinear Schrodinger equations,and obtain the expression of the N-soliton and N-positon solutions.The generating mechanism for two-positon is discussed.Furthermore,the dynamical properties of the smooth positons of the two-component nonlinear Schrodinger are analyzed by using the method of decomposition of the modulus square.In addition,the collision between soliton and positon,and mutual collision of two two-positons are discussed.In Chapter 4,we derive explicit representations of the Darboux transformation and degenerate Darboux transformation for the three-component KunduEckhaus equations.As applications of the Darboux transformation,the abundant and novel interactions between soliton and positon are discussed in detail,such as elastic interactions,inelastic interactions and bound states.In Chapter 5,the conclusion and prospect are presented.