The Analytic Solutions Of The Cauchy Problem For Some Nonlinear Dispersive Wave Equations

The analytic solutions for the nonlinear dispersive wave equations have al-ways been an important branch of nonlinear science, which can make us under-stand the nature of the equations detailedly. Hence, the proof of the analytic solutions for the nonlinear dispersive wave equations have been explored and improved constantly. Among them, applying the classical Cauchy-Kowalevski theorem to proving the analyticity is the most effective method. However, this theorem has certain limitation, because it can only solve the analytical problem of the quasilincar partial differential equations, in which the initial date must be non characteristic. Then the classical Cauchy-Kowalevski theorem has been improved. At last, it is evolved into the abstract Cauchy-Kowalevski theorem. In this thesis, we will use the abstract Cauchy-Kowalevski theorem to prove the analytic solutions of the Cauchy problem for some nonlinear dispersive wave equations.The content of this paper is divided into the following four parts:In first chapter, we briefly introduce the background of this paper, signifi-cance of the research and the progress of the analyticity is also mentioned.In the second chapter, the related theoretical knowledge and definition re-quired to solve the problems in this paper are concretely described.In the third chapter, the analytic solutions of specific equations are dis-cussed, such as a coupled two-component Camassa-Holm system.In the fourth chapter, the analytic solutions of the Cauchy problems for a class of third order nonlinear dispersive wave equations are discussed.