The Traveling Wave Solutions And Their Bifurcations For Several Kinds Of Nonlinear Equations

A lot of phenomena in nature can be described by nonlinear equations.In this doctorial dissertation,based on special structure of equations we investigate the travelling wave solutions and their bifurcations for several important nonlinear wave equations put forward by predecessors through bifurcation method of dynamical system,Hirota bilinear method and limit analysis.The main work of this dissertation is as follows.In Chapter 1,we introduce the background of related issues,research developments,and basic knowledge of our research objects.Furthermore,we briefly summarize the main work of this dissertation.In Chapter 2,for the well-known generalized b-equation,we use the bifurcation phase portraits cleverly to systematically discuss the periodic wave solutions and their bifurcation of the parameter b in different intervals.for b ?(-?,?)and b ?=-1,-2,we get the following results: By selecting the appropriate variable ? as the bifurcation parameter,we get four bifurcation values for ? and reveal two important bifurcation phenomena.The first bifurcation phenomenon is that the elliptic periodic blow-up wave solution can bifurcate into three types of traveling wave solutions,which are trigonometric periodic blow-up wave solution,single blow-up wave solution and smooth solitary wave solution.The second bifurcation phenomenon is that the elliptic smooth periodic wave solution can bifurcate into smooth solitary wave solution.Our work has expanded the previous research results.In Chapter 3,we investigate the new bifurcations of nonlinear waves described by the Gardner equation.We obtain some results as follows: for arbitrary given parameters b and ?,we choose the parameter a as bifurcation parameter.Through the phase analysis and explicit expressions of some nonlinear waves,we reveal two kinds of important bifurcation phenomena.The first phenomenon is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively.The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.In Chapter 4,we innovatively employ the simplified Hirota bilinear method and the bifurcation method to study the traveling wave solutions with arbitrary differentiable functions for two(3+1)-dimensional equations.Firstly,using the simplified Hirota's method,we present generalized multiple soliton solutions and generalized multiple singular soliton solutions in which some differentiable arbitrary functions are involved.Secondly,by means of some special orbits of the traveling wave system and integrating approach,we obtain some other nonlinear wave solutions which also include differentiable arbitrary functions.Our work extends the results of the predecessors.In Chapter 5,a(2 + 1)-dimensional generalized shallow water wave equation is investigated through Hirota bilinear method.Interestingly,the breather type and lump type soliton solutions are obtained.Furthermore,dynamic properties of the soliton waves are revealed by means of the limit analysis.Based on Hirota bilinear method and Riemann theta function,we use some clever analytical techniques for the first time to successfully construct quasi-periodic wave solutions containing arbitrary differentiable functions.We also display the limit properties of these generalized quasi-periodic wave solutions and point out the relation between these generalized quasi-periodic wave solutions and the generalized soliton solutions.The results of the predecessors are expanded.In Chapter 6,we study one of the(3+1)-dimensional water wave equation in Chapter4 once again.Based on the Bell polynomials,we obtain its Hirota bilinear equation which enable us to acquire Riemann theta solutions including arbitrary differentiable functions.Furthermore,we analyze the limit property of these solutions and reveal the relationship between these solutions and the generalized soliton solutions obtained in Chapter 4.Finally,it is a short summary for the work of this paper and the prospect of future research work.