Quasi-periodic Solution And Asymptotic Behavior For Camassa-Holm Equation

In the past decades of years, there has been much interest in investigating the shallow water wave Camassa-Holm equation. Different methods have been used to research this model, including Backlund transformation, inverse scattering method, reciprocal trans-formation, bilinear theory. People have obtained various solutions of CH equation in succession, such as peakon solution, soliton solution, periodic solution, quasi-periodic so-lution and so on, of which we are most interested in the quasi-periodic solution. The one-periodic solution of CH equation has already been obtained in [1], in this paper, we will construct its two-periodic solution based on the results obtained in [1], the main contents are:In chapter1, we will review the basic theory of CH equation and different works which have been done by previous researchers, introduce the quasi-periodic solution of nonlinear integrable equation.In chapter2, the basic knowledge will be introduced in this part, including Riemann theta function and quasi-periodic function. The relation between the periodic solution and quasi-periodic solution will also be refereed in this chapter.In chapter3, we will construct the coupled bilinear form of CH equation, obtain the quasi-periodic solution based on the bilinear form.In chapter4, the asymptotic behavior of the quasi-periodic solution will be ana-lyzed and we will finally prove that the quasi-periodic solution can degenerate into its corresponding soliton solution.In chapter5, the summarization of this paper will be given in this part, we will also give our prospect to CH equation and quasi-periodic solution.