Research On The Dynamical Behavior Of Travelling Wave Solutions For Several Types Of Nonlinear Wave Equations

Nonlinear wave equations are important mathematical models for describing nat-ural phenomena and are one of the forefront topics in the studies of nonlinear math-ematical physics, especially in the studies of soliton theory. The research on findingand analyzing exact solutions of nonlinear wave equations can help us understand themotion laws of the nonlinear systems under the nonlinear interactions, explain thecorresponding natural phenomena reasonably, describe the essential properties of thenonlinear systems more deeply, and greatly promote the development of engineeringtechnology and related subjects such as physics, mechanics and applied mathematics.With the boom of nonlinear science, many models in physics, chemistry and lifesciences can be converted into nonlinear equations, such as nonlinear ordinary dif-ferential equation and partial di?erential equation. Consequently, solving nonlinearequations has become an important research topic in the field of nonlinear science.In this thesis, we investigate several types of nonlinear wave equations by usingthe technique of integral method and bifurcation theory of planar dynamical systems,calculate their travelling wave solutions, and further study on the existence of solitarywave solutions of perturbed nonlinear wave equation. This paper is formed by sixchapters.In Chapter 1, the historical background, research developments and significanceof nonlinear wave equations are summarized.In Chapter 2, the basic theory and method of nonlinear wave equation are pre-sented.In Chapter 3, by using the technique of integral factors, the peakons, solitarypatterns and periodic solutions of generalized Camassa-Holm equation and generalizedG-P equation are obtained.In Chapter 4, we investigate the generalized double sinh-Gordon equation and(N+1)-dimensional sine-cosine-Gordon equation by using the bifurcation theory of pla-nar dynamical systems, discuss and analyze their phase portrait and branches, then,work out the exact solutions of the equation.In Chapter 5, by taking advantage of singular perturbation theory, the existenceof solitary wave solutions of perturbed mKdV equation is proved.Lastly, a summarization of the whole paper and the still unsolved problems in theresearch are given. Moreover, the future study is prospected.