Some Nonlinear Wave Solutions For The Generalized Fornberg-Whitham Equation

In this paper,the qualitative theory of differential equations and the bifurcation method of dynamical systems are used to study the nonlinear wave solutions of the generalized Fornberg-Whitham equation.Under the condition of specific parameters,we respectively obtained bifurcation of phase portraits and precise nonlinear wave solutions of the equation which satisfied n = 2,3,and we also did a research for the case of high degree.The existence of parameter and high order situation brought challenges to the research,we transformed the singular traveling wave system into a regular system by using appropriate traveling wave transformation and time scale transformation,so the singular problem have been solved.Next,we studied the bifurcation of phase portrait of the regular system and obtained explicit nonlinear wave solutions.The main results are as follows:The first chapter is the introduction,which includes three parts.The first is the relevant research background and research significance of this subject,then we introduced the research status of this topic at home and abroad,and stated the academic achievements of many predecessors in this field,finally we summarized the main results.In chapter 2 and chapter 3,we studied the equation with degree n = 2,3 and fond the relationships between the vector fields of the singular traveling wave system and the regular system.By analyzing the singularity and bifurcation curves of the regular system,the bifurcation of phase portraits of the regular system are obtained in the range of different parameters,and the numerical simulation is carried out respectively.Finally,when n = 2,the expressions of nonlinear wave solutions of the equation,such as solitary wave solutions and peak solitary wave solutions,are obtained under certain circumstances.When n = 3,the explicit expressions of one peak solitary wave solutions of the equation are obtained,and the correctness is verified by code.In the fourth chapter,according to the previous two chapters,we did a research for the case that the degree n is a general positive integer.By numerical analysis of n = 4,6,it is found that the number of singular points and the bifurcation of the regular system are similar to the case whose degree is 2 when degree n is even.Similarly,by numerical analysis of n = 5,7,it is found that the case of the degree which is odd is similar to the case of n = 3.