Integrable Extensions For Camassa-Holm Type Equations And The Matrix (2+1)-dimensional Integrable System

Integrable systems with self-consistent sources (with sources) have wide appli-cations in mathematical and physical fields. In this thesis, we construct Camassa-Holm (CH) equation with sources, Degasperis-Procesi (DP) equation with sources,2-component Camassa-Holm (2-CH) equation with sources and the new extended ma-trix Kadomtsev-Petviashvili (KP) hierarchy, obtain their Lax representations and studytheir solutions. By adding variation derivative of the eigenvalue to the potential in theHamiltonian structure of the corresponding integrable systems, we introduce the CH,2-CH and DP equations with self-consistent sources, and derive their Lax representa-tion. With the square eigenfunction symmetry constraint, we add new time seriousτBin the matrix KP hierarchy to obtain new extended matrix KP hierarchy. The extendedmatrix KP hierarchy contains two time series tA andτB and eigenfunctions and adjointeigenfunctions as components. The 2+1-dimensional AKNS equation with sourcesand the Davey-Stewartson (DS) equation with sources are obtained. The second typeof 2+1 AKNS equation with self-consistent sources and the second type of DS equa-tion with self-consistent sources are fistly constructed. The new extended matrix KPhierarchy and its tA andτB reductions gives a simple way to construct two types of 2+1and 1+1 dimensional functions with self-consistent sources.CH, DP and 2-CH equations have been studied from many points of view. Wehave relations between those equations and some hierarchy's first negative ?ow by re-ciprocal transformations. In this thesis, we firstly construct the generalized reciprocaltransformation for CH, DP and 2-CH equations with self-consistent sources. By com-bining method of variation of constants and Darboux transformation method, we obtaindi?erent types of solutions for CH, DP and 2CH equations with sources, containingsoliton, peakon, cuspon, positon and negaton.Dressing method is an important method in solving the KP equation. However thedressing method for the matrix KP hierarchy can not be directly applied to the extended matrix KP hierarchy. We propose a generalized dressing method for extended matrixKP hierarchy and obtain some of its solutions. This method provides a simple andnatural way to obtain the solution of the new extended matrix KP hierarchy. In thisthesis, we give the soliton solutions of two types of 2+1 dimensional AKNS equationwith sources and two types of DS equation with sources.