Orbital Stability Of Elliptic Periodic Peakon Of One MCH Equation

In this paper,the bifurcation theory of planar dynamical systems is used to study the exact parameter expression of the traveling wave solutions of the modified Camassa-Holm(mCH)equation and the orbital stability of the elliptical periodic peakon.First,we convert the modified Camassa-Holm equation into a planar system and then we obtain the first integral and algebraic curves of this system.Based on the first integral and algebraic curves,we obtain a new peakon with hyperbolic function representation.Moreover,some new periodic peakon are given by elliptic functions and triangle functions.For the orbital stability of the elliptic periodic peakon of this equation,we mainly use the theory of elliptic function and elliptic integral is used in the calculation,by using the invariants of the equation and controlling the extrema of the solution,it is demonstrated that the shapes of these elliptic periodic peakon are stable under small perturbations in the energy space.To the best of our knowledge,this is the first result on the orbital stability of elliptic periodic peakon.In this paper,we will divide into five parts,and the main framework arrangement is as follows:In Chapter 1,we mainly introduce the study background,study status and main research contents of the modified Camassa-Holm equation.In Chapter 2,the bifurcation theory of traveling wave solutions of nonlinear wave equations and the related knowledge of elliptic integral and elliptic function are introduced.In Chapter 3,The traveling wave solution of the modified Camassa-Holm equation is discussed using the branching theory of the dynamic system.First,the equation is transformed into a plane system,and then the smooth periodic wave solution and solitary wave represented by the precise parameters are obtained,as well as new peakon and periodic peakon solutions.In Chapter 4,the elliptic function and elliptic integral theory are used to calculate the modified Camassa-Holm equation,and it is proved that the elliptic periodic peakon of the equation is orbitally stable.In Chapter 5,the research content of this article is summarized and looked forward to future research topics.