Persistence Of Solitary And Periodic Waves In Some Kinds Of Shallow Water Equations Under Nonlocal Perturbation

The shallow water wave equation in fluid mechanics has always attract-ed researchers’attention,it is a nonlinear partial differential equation,which describes the motion of waves and tides in oceans and rivers.Traveling wave solution is an important type of solution in nonlinear wave equations,a great number of nonlinear wave equations have been found to have a variety of trav-eling wave solutions.In this paper,the geometric singular perturbation theory,Abelian integral and Melnikov method are combined to study the existence of solitary wave and periodic wave solutions for some kinds of shallow water wave equations under nonlocal perturbation,including solitary waves of the perturbed Kd V equation with nonlocal effects,existence of solitary wave solutions for delayed Fornberg-Whitham equation,solitary wave and periodic wave solutions of ZK-KS equation and solitary wave and periodic wave solutions of generalized ZK-KS equation.There are seven chapters in the whole text,and the main contents are arranged as follows:In chapter 1,we briefly introduce the research background,the research status at home and abroad,and the main contents of this paper.In chapter 2,we introduce some theoretical knowledge needed in this paper:Fenichel theory and Melnikov method.In chapter 3,solitary waves of the perturbed Kd V equation with nonlo-cal effects are studied.First the existence of solitary wave solutions for the original Kd V equation is proved;Then the existence of solitary wave solutions is established for the equation with two types of delay convolution kernels by using the method of dynamical system,especially the geometric singular per-turbation theory,invariant manifold theory and Melnikov method.Moreover,an interesting result is obtained in the case of only delay without backward diffusion perturbation:there is no solitary wave solution in the case of local delay,but there is a solitary wave solution in the case of nonlocal delay;Finally,the asymptotic behaviors of solitary wave solution are discussed by applying the asymptotic theory.In chapter 4,we study the existence of solitary wave solutions for the delayed Fornberg-Whitham equation.Firstly,by applying the phase plane analysis method of the planar dynamic system,some qualitative properties and traveling wave solutions for the equation without delay and perturbation are analyzed in detail.Then the geometric singular perturbation theory is used to analyze the delayed Fornberg-Whitham equation,transform it into the standard form of singular perturbation,and calculate the corresponding Melnikov integral,so as to prove the existence of solitary wave solutions.The delays discussed in this chapter are local delay and nonlocal delay,which are expressed in different convolution forms,and the results obtained are also different.In chapter 5,the existence of solitary wave and periodic wave solutions for the ZK-KS equation is studied.Firstly,solitary wave and periodic wave solutions of the Zakharov-Kuznetsov equation are proved by employing the phase plane analysis method.Then we discuss the existence of solitary wave and periodic wave solutions for the ZK-KS equation by using the geometric singular perturbation theory and the regular perturbation analysis for a Hamil-tonian system.It is proved that the wave speed(8₀（?）is decreasing on?by analyzing the ratio of Abelian integrals,and the upper and lower bounds of the wave speed are obtained.Finally,the persistence of solitary wave solutions for delayed ZK-KS equation is established.In chapter 6,we study the solitary wave and periodic wave solutions of the generalized ZK-KS equation.This chapter generalizes some results obtained in Chapter 5,we find that the results are mainly divided into two cases: nis odd or even.The case of n=1 has been discussed in Chapter 5,and the case of n=2 is discussed at the end of this chapter.However,the monotonicity of the wave speed can’t be obtained for arbitrary integer n≥3,which is a problem to be solved.In chapter 7,we summarize the main works of this paper,and propose some problems to be studied in the future.