Qualitative Analysis And Exact Traveling Wave Solutions For The KG Equation And The Perturbed KG Equation

In this paper,we investigate the Klein-Gordon?KG?equations and the corresponding ones with perturbation terms which have many applications in nonlinear system.The purpose of this dissertation is to study the existence of bounded traveling wave solutions for the KG equations with higher-order nonlinearity and the corresponding ones with perturbation terms,the relation between the periodic solutions and the solitary wave solutions,the evolvement of their solitary wave solutions due to the influence of perturbation,and their solutions.By using the theory of planar dynamical systems,the energy level of traveling wave systems,special function integral method,undetermined coefficients method and the thought of homogeneous principle,we take the following equations as examples to investigate the above problems.?1?The KG equation with cubic nonlinear term u_tt-au_xx?u-?u³ = 0,????2?The perturbed KG equation with cubic nonlinear term u_tt-au_xx+?u-?u³=??bu_t+cu_x?,????3?The KG equation with fifth-order nonlinear term u_tt-au_xx+?u-?u³+?u⁵=0,????4?The perturbed KG equation with fifth-order nonlinear term u_tt-auxx+?u-?u³+?u⁵=??bu_t+cu_x?.???We carry on the study of this paper carefully,from simple to complex,from easy to difficult.And the main results are represented as follows.1.Applying the bifurcation theory of planar dynamical systems,we carry out qualitative analysis for the traveling wave systems corresponding to equations?I?-?IV?,respectively.We obtain the types of finite and infinite singular points for these above systems and depict global phase portraits associated with them under different parameters conditions.Furthermore,we present the existence,the numbers and the probable behaviors of bounded traveling wave solutions for equations?I?-?IV?.2.We obtain all exact explicit expressions of the bounded traveling wave solutions for the equations???and???.This is a difficult problem in the paper.To overcome it,we study energy curves of the traveling wave systems corresponding to the equations?I?and?III?under various parameter conditions.We establish the relation between the bounded traveling wave solution and the energy level h.By using suitable transformation and special function integral method,we obtain bell-shaped solitary wave solutions,kink-shaped solitary wave solutions and periodic wave solutions,and discuss the evolutions of these periodic solutions as the energy level h changes.3.We investigate the perturbation effect on the behavior of the bounded traveling wave solutions of the equations???and???.The results shows,that a bounded traveling wave appears as a kink-shaped solitary wave when ? is more than some critical value,a bounded traveling wave appears as a damped oscillatory wave when ?is less than some critical value,and a bounded traveling wave appears as a non-monotone and non-oscillatory bounded traveling wave when ? belongs to some bounded open interval.4.From the theory of rotation vector field in the planar dynamical systems,we know the saddle-focus orbit or the focus-saddle orbit responding to the damped oscillatory solution of the equations???and???rises from the responding homoclinic orbit or heteroclinic orbit of the equations???and???under the influence of the perturbation terms essentially,respectively.Based on the bell solitary wave solutions and kink profile solitary wave solutions of equations???and???,we design the structure of the damped oscillatory solutions,and obtain the approximate damped oscillatory solutions of the equations???and???by using undetermined coefficients method.5.We analyze the error between the approximate damped oscillatory solutions and exact solutions of equations???and???.This is another difficulty of this paper.In order to solve this problem,we establish the integral equation reflecting the relations between the approximate damped oscillatory solutions obtained in this dissertation and the exact solutions corresponding to them by the idea of homogenization principle,and then obtain their error estimate.We can prove that the error between exact solution and approximate damped oscillatory solution is infinitesimal decreasing in exponential form.The results of this paper can reveal the relationship between periodic wave solution and solitary wave solution of KG equation with higher-order nonlinear terms,as well as the perturbation effect on the solution for the nonlinear system.Therefore,it is meaningful in solitary wave theory and its applications and can be referred to further study the higher-order nonlinear and perturbed system.