On The Matrix Camassa-Holm Equation

In recent years,the Camassa-Holm(CH)equation has attracted much attention and various studies.The most interesting feature of the CH equation is that it admits peaked soliton(peakon)solution.In this paper,we present an sl(2)matrix version of the CH equation.This matrix CH equation allows for an arbitrary matrix function to be involved in.We may generate many integrable peakon systems with different choices of the arbitrary matrix function.We show that the matrix CH equation possesses Lax pair and infinitely many conservation laws.In addition,we take some examples to discuss in detail the N-peakon solutions for some equations containing in the matrix CH equation.The thesis is divided into three chapters.In the second chapter,by introducing a pair of 4 × 4 matrix spectral problems,we provide a Lax pair and construct the infinitely many conservation laws for the matrix CH equation.We also show how to choose the arbitrary matrix function in the matrix CH equation.Based on this,we find a lot of integrable CH-type equations with peakon solutions.In the third chapter,we derive the single peakon solutions and the N-peakon dynamical systems for several new matrix CH equations which are the special cases of the matrix CH equation.