The Source Generalization Of The Camassa-holm Equation And The Noncommutative Integrable Equation And Its Solutions

Soliton and integrable system is an important branch of mathematical physics and applied mathematics.In recent years,the hot spots lie in the multi-component integrable system and the noncommutative system.Some problems are explored in this dissertation,such as the source generalization of CH equations and of noncommutative integrable hierarchies,integrable properties,and how to solve these new integrable systems with sources.There are three parts in this dissertation.The specific results are given as follows:1.The first part focus on the study on source generalization,integrable properties and the solutions of modified Camassa-Holm equation(mCH).Firstly,the mCH eqution with selfconsistent sources(mCHESCS)and its related Lax pairs are derived;Secondly,the infinite conservation laws and reciprocal transformation of the mCHESCS are also constructed.By means of the reciprocal transformation of mCHESCS we presented,together with the Darboux transformation of mCH equation,some new solutions of mCHESCS are finally obtained,such as the multisoliton,multinegaton and multipositon solutions.2.The second part mainly aims at the source generalization of the two-component modified Camassa-Holm equation and its integrable properties.To begin with,the two component mCH eqution with self-consistent sources(2-mCHESCS)and its Lax pairs are derived;Moreover,The infinite conservation laws and reciprocal transformation of the 2-mCHESCS are also constructed.3.The third part focus on the source generalization of noncommutative modified KP(ncmKP)hierarchy and its solutions.The noncommutative modified KP hierarchy with self-consistent sources(ncmKPHSCS)is firstly constructed,which gives rise to two types of ncmKP equation with self-consistent sources(ncmKPESCS).Then,the noncommutative Miura transformation of between ncKPHSCS and ncmKPHSCS is presented,and the quasideterminant solutions of the ncmKPHSCS are also given.As its byproduct,the quasideterminant solutions of two types of ncmKPESCS are obtained.The matrix solutions of two types of ncmKPESCS are finally investigated,and the impact of noncommutativity on the soliton equation with sources is analyzed.