The Existence And Asymptotic Behavior Of Traveling Wave Solutions For Two Types Of Evolution Equations

Traveling wave solution is one important type of solution to the evolution equation,which describes the evolution laws in physical,biological,chemical et al.In order to study such laws,one must firstly find the traveling wave solutions.However,it is sometime difficult to work out them because of the complex of the equation itself and the limitation of the traditional solving methods.Therefore,it is necessary and of theoretical value to study the traveling wave solution of evolution equation from the perspective of theoretical analysis.Firstly,in chapter one,the research background of evolution equations are introduced.Then we show the relevant literature at home and abroad.Finally,the significance of the research is given and contents of each chapter are listed.The second chapter is concerned with the existence of solitary wave solutions to a singularly perturbed generalized Gardner equation with nonlinear terms of any order.Based on the relation between solitary wave solution and homoclinic orbits which are of the associated ordinary differential equations,when the perturbation parameter is sufficiently small,we use geometric singular perturbation theory to obtain the persistence of solitary wave solutions for this equation.The numerical simulations can verify our theoretical analysis.In the third chapter,we study the existence and asymptotic behavior of traveling wave fronts for a food-limited population model with spatio-temporal delay.Using geometric singular perturbation theory and Fredholm alternative,we establish the existence of traveling wave fronts of this model for sufficiently small time delay.The approach is to reformulate the problem as an existence question for a heteroclinic connection inR~6.Furthermore,employing standard asymptotic theory,we obtain the asymptotic behavior of traveling wave fronts of this model for the first time.Finally,chapter four summarizes the full text.