Analyticity Of The Cauchy Problem For A Class Of Nonlinear Evolution Equations

In this thesis,we consider the analyticity of the solutions to the Cauchy problem for the integrable modified Camassa-Holm equation,the two-component Novikov system and the higher-order Camassa-Holm equation.Ap-plying the classical Cauchy-Kowalevski theorem to proving the analyticity of the solutions to the Cauchy problem for some equations is the most effective method,using this theorem has certain limitation.With the further of study,the classical Cauchy-Kowalevski theorem has been changed into the abstract Cauchy-Kowalevski theorem.In this thesis,firstly,we establish the analyticity of the solutions to the Cauchy problem for the integrable modified Camassa-Holm equation and the two-component Novikov system by using the abstract Cauchy-Kowalevski theorem.Then,we give and apply the Fourier transforma-tion to prove a lemma,on the basic of the lemma,we use the abstract Cauchy-Kowalevski theorem to establish the analytic solutions of the Cauchy problem for the higher-order Camassa-Holm equation.The following is the arrangement of the content of this thesis:At first,we briefly describe the background of the analyticity,all kinds of methods and the progress of the analyticity.Secondly,we list the related definitions and theories in order to establish the analyticity.Thirdly,we obtain the analyticity of the solutions to the Cauchy problem for the integrable modified Camassa-Holm equation and the two-component Novikov system.Fourth,we obtain the analyticity of the solutions to the Cauchy problem for the higher-order Camassa-Holm equation.