Branched Traveling Wave Solutions Of The Generalized Camassa-Holm Equatio

Nonlinear partial differential equation is an important branch of modern mathematics.It is often used to solve problems in mechanics,economics,epidemiology,ecology,and so on.The Camassa-Holm equation occupies an important place in nonlinear partial differential equation.It is a shallow water wave equation.In recent years,experts have studied it in many aspects.Solving nonlinear partial differential equation is a popular research direction.In this paper,by using the bifurcation theory of planar dynamical systems to study the traveling wave solutions of generalized Camassa-Holm equation and the generalized Camassa-Holm equation with dual-power law nonlinearity.First,by analyzing the bifurcation curves that varie with parameters.Next,the phase diagrams with different parameters are obtained using mathematica to explore the orbits formed by different singular points,and the orbital equations corresponding to the different orbits are analyzed to obtain the exact traveling wave solutions of the two generalized Camassa-Holm equations by integration.The structure of this article is as follows,In Chapter 1,we briefly introduce some background,including the origin and development of solitary waves,the current status of the Camassa-Holm equation,and the methods for solving nonlinear partial differential equation.In Chapter 2,we study the generalized Camassa-Holm equation with dual-power law nonlinearity ut-μuxxt+2κux+ηuxxx=f(u)ux+s(2uxuxx+uuxxx),(1)where f(u)=Au+Bum,when m=1.m=2,by dividing the parameters into different cases,the bifurcation diagram problem of equation(1)is discussed by using the bifurcation theory of dynamical system,in order to obtain the specific expressions of traveling wave solutions through integration,such as solitary wave solutions,periodic wave solutions,peakon,singular wave solutions and so on.In Chapter 3,we study the generalized Camassa-Holm equation,when p=2,p=3,by discussing the transformation of parameters,the bifurcation diagram problem of equation(2)is discussed by using the bifurcation theory of dynamical system,in order to obtain the specific expressions of traveling wave solutions through integration,such as peakon,solitary wave solutions and so on.In Chapter 4,we summarize the work of this paper,and the issues that can be further investigated.