The Existence Of Solitary Wave Solutions Of Viscous Camassa-Holm Equation

In recent decades,great achievements have been made in the study of nonlinear shallow water wave equations.Among them,the research on the landmark Camassa-Holm equa-tion has attracted many experts and scholars.Different from the traditional research on CH equation,this paper studies the Camassa-Holm equation with viscous term,also known as Navier-Stokes-alpha equation.We study the existence of solitary wave solutions under two different perturbations.One is CH equation with viscous term,another is CH equation with non-Newtonian fluid properties.In this paper,according to the relationship between solitary wave and homoclinic orbit,the partial differential equation is transformed into the ordinary differential equations with slow-fast variables.The existence of solitary wave solutions is proved by using the dynamic system method,especially the geometric singular perturbation theory,then combining with the phase diagram analysis.The necessary and sufficient con-ditions for the existence of periodic orbits and unique homoclinic orbits are given.When we analyze the existence of solitary wave solutions and periodic solutions,we discuss them in two cases: the case of a = 0and the case of a = 0respectively.Finally,we find that although the solitary wave solutions of CH equation with viscous term exists,that of the perturbed CH equation with non-Newtonian fluid property does not exist.