The Nonlinear Wave Solutions And Bifurcations Of The High Order B-equation

In this paper,we systematically investigate the traveling wave solutions and bifur-cations of the n degree generalized b equation by using the bifurcation theory in the dynamic system,and obtain the qualitative behaviors and the expressions of traveling wave solutions of the equation in the three cases of b=0,b>1,and b<-1,respectively.The main difficulty in analyzing such equations is that the equations have high nonlinear convective terms,which requires more theoretical analysis and numerical computation.It is also difficult to obtain nonlinear wave solutions,bifurcation parameters and bifurcation curves of the equations in the high case,and how to deal with the singularity of the cor-responding traveling wave system.We transform the singular traveling wave system into a regular system using appropriate traveling wave transformation and time scale trans-formation.After that the vector fields and bifurcation of regular systems are investigated by means of the bifurcation theory in dynamical system and numerical methods.Based on the transformation and the properties of the regular system,the phase track of the singular travel wave system are obtained.Finally,we can discus the bifurcation behavior and dynamics of the nonlinear wave solutions of the equation by phase orbits,and give the expressions for these solutions.The main results of this paper are as follows.1.When b=0,the relationship between the singular traveling wave system and the regular traveling wave system is obtained.The phase portrait analysis reveals some new phenomena,such as an infinite number of closed orbits crossing a singular line and intersecting at two points in the traveling wave system under certain parameters;some homoclinic orbits have no singular point inside,and so on.The theoretical analysis reveals that there are three kinds of bifurcation phenomena in the equation,including the bifurcation of solitary wave and periodic wave,solitary wave and blow-up wave and double solitary wave.2.When b>1,k=0,the bifurcation parameters and bifurcation curves of the traveling wave system are numerically determined,and the bifurcation wave velocity of the peakon solution and the anti-peakon solution is obtained,as well as the maximum wave velocity of the peakon solution if n is even.Under different parameter conditions,the phase portraits of the regular traveling wave system are established,it further reveals the relationship between various nonlinear wave solutions,and extends and improves some of the previous results.3.By using the dynamic system bifurcation method,we can obtain the expressions for multiple nonlinear wave solutions in the three cases of b=0,b>1,and b<-1,respectively.When b=0,the parameter conditions and expressions for the existence of solitary wave solutions,periodic wave solutions and bursting wave solutions are given;when b>1,we obtain the explicit or implicit expressions for peakon solutions,anti-peakon solutions,smooth solitary wave solutions and smooth periodic wave solutions;when b<-1,the parameter conditions for the existence of periodic wave solutions and their implicit expressions are given.This paper is divided into five parts.The first chapter is an introduction,which summarizes the history of nonlinear wave equations and soliton,the current state of research,the main research methods and results,and a brief description of the main contents of this paper.It also introduced the theories and methods used in this paper.The second to fourth chapters investigate the nonlinear wave solutions and branching problems of the high-order b equation in the three cases of b=0,b>1,and b<-1,respectively,and reveal the relationship between multiple nonlinear wave solutions.The last chapter summarizes the results of this paper and raises questions that need to be explored further.