Research On Persistence Of Traveling Wave Solutions For Two Kinds Of Nonlinear Evolution Equations

In this paper,the persistence problems of travelling wave solutions for two kinds of nonlinear evolution equations are studied by using geometric singular perturbation theory and Melnikov function method.The main research contents are as follows:Firstly,the travelling wave solution of nonlinear evolution equation and its related background knowledge,the work done by predecessors and the main research content and results of this paper are introduced.Then,we study the persistence of the solitary wave solution of the generalized Klein-Gordon equation.Firstly,the three-dimensional problem is transformed into two-dimensional problem by using the theory of singular perturbation.Secondly,using Melnikov function method,it is proved that the homoclinic orbit of the generalized Klein-Gordon equation persists under the condition that the parameters are small enough.Finally,the persistence of the wave front solution of Van der Waals equation with fifth order dispersion term is studied.Since the wave-front solution corresponds to the heteroclinic orbit in three-dimensional space,the dimension of the stable and unstable manifolds of its equilibrium point is calculated by using the Argument Principle.Secondly,since the existence of heteroclinic orbits in three-dimensional space is a difficult problem,the critical manifold of slow system is proved to be normal hyperbolic by using the theory of geometric singular perturbation,and then the three-dimensional problem is transformed into a two-dimensional problem.Finally,the implicit function theorem is used to prove the persistence of heteroclinic orbits in undisturbed systems.Then,Matlab software is used for numerical simulation to verify the validity and rationality of the previous proof.