Orbital Stability Of Periodic Peakons For Two Kinds Of Nonlinear Camassa-holm Equations

Camassa-Holm type equation is a very important kind of shallow water wave equation,which has a wide range of application backgrounds.The equation has continuous but not smooth solitary wave solutions,and it is completely integrable.The main work of this thesis consists of the following two parts.In the first part,the orbital stability of the periodic peakon of the higher-order nonlinear Camassa-Holm equation and the existence of the periodic 2-peakon are proved.Firstly,a global identity about conserved quantity is established.Secondly,the maximum perturbation term u(t,x)is determined by constructing an auxiliary function F(M,m),and the global perturbation between the maximum value of the solitary wave generated by the perturbation near the periodic peakon and the maximum value of the periodic peakon is estimated.Finally,the periodic peakon is proved to be orbital stable,and the existence of periodic 2-peakon is proved by using the method in reference [56].In the second part,the existence of periodic peakon to the generalized modified Camassa-Holm equation is first verified,and then the orbital stability of periodic peakon is proved by using the extremum determination of bivariate functions.In Chapter 1,we introduce the significance and current research status of the orbital stability of periodic peakons for Camassa-Holm type equations,as well as the main work of this thesis.In Chapter 2,the orbital stability of periodic peakon and the existence of periodic2-peakon for a high-order nonlinear Camassa-Holm equation are proved.In Chapter 3,the proof of the orbital stability of the periodic peakon for the generalized modified Camassa-Holm equation is given.In Chapter 4,a summary and outlook of the work in this paper are presented.