Existence Of Solitary Wave Solutions For The Third-order Nonlinear Dispersive Partial Differential Equations And Traveling Wave Solutions For KdV Equations

The family of third-order nonlinear dispersive partial differential equations is a class of nonlinear nonlinear partial differential equations.It satisfies the necessary con-dition of symmetry integrability and complete integrability,Moreover,three famous shallow water wave equations can be obtained by different transformations:KdV e-quation?Camassa-Holm and Degasperis-Procesi equation.KdV equation has been applied in various fields of physics.In recent years,it has important significance for the study of delayed KdV equations.In this paper,by using the method of dynamic-s system,especially geometric singular perturbation theory,combined with invariant manifold theory,implicit function theorem and the Abelian integral,we study the trav-eling wave solutions and solitary wave solutions of the perturbed third-order nonlinear dispersion partial differential equation and perturbed generalized KdV equation with distributed delay.This paper includes four chapters as follows:In chapter 1,we briefly introduce the backgrounds,significance and the main con-tent of this paper.In chapter 2,we introduce some basic concepts and preliminary knowledge.In chapter 3,we study the solitary wave solutions of the perturbed third-order nonlinear dispersion partial differential equation.We found that there are three equa-tions that satisfy the integrability condition within this family:the KdV equation,the Camassa-Holm equation and the so-called Degasperis-Procesi equation.Firstly,the Hamiltonian function was constructed,and the existence of solitary wave solutions for undisturbed equation was proved by looking for the Homoclinic orbits.Secondly,by using the GSPT and the IMT,the existence of solitary wave solutions for the per-turbed third-order nonlinear dispersion partial differential equation is proved by Mel-nikov function discriminant method.In chapter 4,we study the solitary waves and periodic waves of a generalized KdV equation with distributed delay.It has a wide range of applications in physics and fluid mechanics,and describes shallow water waves with weak nonlinear restoring forces.By using the GSPT and the IMT,we can prove the existence of solitary wave solutions and periodic wave solutions for a generalized KdV equation with distributed delay.The range of the limit wave velocity and the monotonicity can be obtained by the calculation of Abelian integral.