A Class Of Analytic And Persistent Development Of Nonlinear Equations

In this thesis, we consider the analytic solutions of the Cauchy problem for a kind of nonlinear partial differential equations, which can rewrite in the following form:We know that the methods of proving the analytic solutions for the nonlinear partial differential equations(PDE) have been explored consistently, which now has become a main part of the nonlinear science. As is known to all, the classical Cauchy-Kovalevskaya theorem is the typical method. However, in this thesis,we emphatically introduce the abstract Cauchy-Kovalevskaya theorem,which is ob-tained by predecessors recently. Meanwhile, we apply this theorem to prove the analytic solutions of some shallow water equations. In addition, the persistence property of these equations is also discussed.The thesis is arranged as follows:At first, we briefly present the concepts and the research value of the ana-lyticity and the persistence property. In addition, the progress of the proof ways for this two properties is also mentioned.Secondly, the preliminary theories are introduced in detail. In the first place, we demonstrate the theory of establishing the analytic solutions, and the relationship between the two kinds of ways of proving the analyticity of the solutions for nonlinear PDE is displayed. Next, we list the preliminary of the persistence property.At last, the analytic solutions of some kinds of nonlinear shallow water e-quations are discussed. Firstly, we consider the analytic solutions of the Cauchy problem for Dullin-Gottwald-Holm(DGH) equation; Secondly, the analytic solu-tions of the Novikov equation are investigated; Finally, we deal with the analyt-icity and the persistence property of the Fornberg-Whitham equation.