The Persistence Of Traveling Wave Solutions Of A Singularly Perturbed Nonlinear Wave Equation

Traveling wave solution is a kind of similarity solution,which is widely existed in all kinds of nonlinear equations,and its typical characteristic is that this kind of solution can keep the translation invariant in space.Many phenomena in physical,chemical and biological can be described by nonlinear equations,such as shallow water wave motions in fluid dynamics,ion acoustic waves in pleasmas and traffic flows,etc.Understanding and discovering the intrinsic mechanism of these nonlinear equations has become an important research topic in the field of nonlinear wave theory and numerical analysis.In the study of nonlinear science,the theory and method of dynamical system have become one of the active frontier problems in nonlinear science due to its profundity and wide application.Therefore,it is very promising to apply the theory and method of dynamical system to the study of nonlinear wave equation.In recent years,the study of geometric singular perturbation theory and its appli-cation have made great progress on the whole world.Geometric singular perturbation has become a hot topic and research field concerned by many scholars.So far,singular perturbation theory is still an effective theoretical method to solve the weak nonlinear problem in mechanical,acoustic,atmospheric ocean and engineering.In this paper,we study perturbed BBM equation and Zakharov-Rubenchik equation.For perturbed BBM equation,after traveling wave translation and integration,it becomes an ordinary differential equation of third order singular perturbation.Then we investi-gate the bifurcations,phase portraits and exact traveling wave solutions of the equation when the small parameter ε is zero,and the persistence of traveling wave solutions of the equation when ε>0.To be specific,firstly,we get the equivalent planar dynamic system when ε=0.Secondly,using the dynamical system and the bifurcation theory,we study the bifurcations and phase portraits of the system when the integral constant is different.Then we use the method of elliptic integral function to find the exact traveling wave solutions.Finally,the perturbed BBM equation is reduced to a regular pertur-bation system by using the geometric singular perturbation theory,and the existence of homoclinic orbits and periodic orbits is discussed by using the method of successive functions.For Zakharov-Rubenchik equation,using the traveling wave transformations and integrating it once,Zakharov-Rubenchik equation becomes an ordinary differential equation.we derive the equivalent planar dynamic system.Then the type of equilib-rium points of the Zakharov-Rubenchik equation are obtained when the coefficients are different and the phase portraits are given.Finally,we give partial exact traveling wave solutions of Zakharov-Rubenchik equation.