Multi-component Camassa - Holm Equation Is A Kind Of New Form

Russell first observed shallow water in British mechanics Lab from the1983. Different meaning of shallow water equations will get in different conditions.For instance, Kdv equation will get in long wave, small amplitude condition.In the completely integrable system, the shallow water of the partial differential equa-tion qualitative theory of the research has become known as important research topic in the mathematics and physics.Therefore, the completely integrable shal-low water wave equation with the very important research value. Camassa and Holm is deduced the other kind of shallow water equations in the United States.That is Camassa-Holm equation.This paper mainly studies the integrable system of two component Camassa-Holm equation with a new form solution and extended to multiple components of the case. The paper is organized as follows:In chapter1, describes the soliton peak blow-up phenomenon, Hamiltonian structure and H1norm conservation law of the Camassa-holm equation.Gives the current situation of the research background development and its applica-tions The main content of this paper is presented.In chapter2,base on Robert Camassa and Darryl found the diffusion of the shallow water equations is completely integrated in1993:Camassa-holm equa-tion. First of all, discussed the two components Camassa-holm equations with soliton peak form solution. Obtained two components Camassa-holm equations has indeed two soliton peak form solution.In chapter3, studied the three components Camassa-holm equations with peak form solution. Gained three components Camassa-holm equation has indeed two soliton peak form solution.Then extended two components of the Camassa-holm equation to N components of also has two peak form soliton so-lution. Finally extended two soliton peak form solution to N soliton peak form solution.