Researches On Traveling Wave Solutions Of A Class Of Nonlinear Wave Equations

Nonlinearity is universal and important phenomenon in nature. Nonlinear Science, which has soliton, fractal and chaos theories as its main parts, is the subject of studying the nonlinearity. Most nonlinear problems can be described by nonlinear equations.In the nonlinear systems, the soliton theory of the nonlinear wave equations is the important topic. The key problem in soliton theory is to get solutions of the nonlinear equations. As a leading subject and hot interest in nonlinear science, study on the solution method of the nonlinear wave equations has become more and more challenging. At present, although a number of methods are proposed and developed to look for the exact solutions of the nonlinear wave equations, unfortunately, not all these approaches are universally applicable for solving all kinds of nonlinear wave equations directly.As a consequence, it is still a very significant task to go on searching for various powerful and efficient approaches to solve nonlinear wave equations.This dissertation is based on systematic research and on the existing technique of solving nonlinear wave equations.we do more meticulous research on traveling wave solutions of a class of nonlinear wave equation with the physical context ,from the qualitative and quantitative point of view.The studies enrich and develop the contents of the nonlinear wave equations ,are of some theoretical significance and application value.This dissertation consists of eight chapters. In Chapter 1 and Chapter 2, we introduce the historical background, study development of nonlinear wave equation and several important nonlinear wave equations . The methods known up to today for solving the nonlinear wave equation are summarized and analyzed. Then the concerned concepts and theories which used in this paper are introduced and the primary contents of this dissertation are reported as well.In Chapter 3, the travelling wave solutions for BBM-like equations with fully dispersion are studied. By introducing the concept of nonlinear intensity, some classical methods are employed to study solutions of nonlinear wave equation with more complex nonlinear terms (BBM-like equations with fully dispersion) and find abundant solitary wave solutions:solitary pattern solutions expressed in terms of the hyperbolic sine, cosine and tangent functions,smooth solitary wave solution , kink solution, anti-kink solution, floating solitary wave solution, peakon solution .By the auxiliary equation method,We establish the relationship between a form of solitary wave solutions and P(u), obtain the peakon solution and the singular solitary wave solution.Finally, we give an analysis of P(u) as well as the coefficient on forms of the solutions.the traveling wave solutions of ZK-BBM equation and general BBM equation are investigated by qualitative analysis method. By studying the bifurcation of this equation and dynamic characteristics, we give the expressions of the solitary wave solution and periodic solitary wave solution according with the bifurcation theory. The limit of periodic solitary cusp wave solution and solitary wave solution both equal to the peakon solution. Then the expressions of the solitary wave solution and periodic solitary wave solution are given at various parameters conditions. Some figures are presented by numerical simulation.In Chapter 4, new Miura type transformation between nonlinear dis- persive wave equations is established. A new algebraic method is devised to construct a transformation relating the complicated nonlinear wave equations with the simpler ones. A characteristic feature of our method lies in that the travelling wave solutions of an aimed equation can be obtained by the solutions of a simpler equation directly. Another characteristic feature is that conditions under which different solutions appear can be given. We choose the nonlinear dispersive generalized KdV equation (K(m+1,2)) and the mKdV equations to illustrate our method. As a result, abundant travelling wave solutions of the K(m+1,2) equation are obtained, including periodic solutions, smooth solutions with decay, solitary solutions and kink solutions.In Chapter 5, the b-family of the modified DP-CH equation are discussed. By introducing a parameter b and using the extended tanh method , the rational hyperbolic functions method and the rational exponential functions method,We extend the solutions of a class of wave equation. Not only some are in very good agreement with those obtained in some literatures, but also the conclusion is more general.In Chapter 6, application of the F-expansion methed are studied. Using the generalized F-expansion methed, solitary wave solutions of Mizhnik-Novikov-Veselov equation, Klein-Gordon equation, Modified Benjamin-Bona-Mahony equation are obtained.In Chapter 7, some form of solitary wave solutions in the actual application is investigated.The last chapter, Chapter 8, is devoted to concluding with a short summary and further commentary.