Bifurcation Of Traveling Wave Solutions In A Generalized RLW Equation

In the view of nonlinear science, a lot of practical physics, chemistry, and life sciences models can be changed into nonlinear equations (such as nonlinear ODE, PDE and difference equation). It becomes an important topic for nonlinear scientists to study the finite travelling wave solutions of a nonlinear equation. To find exact travelling wave solutions for a given nonlinear system, a lot of methods have been developed such as the inverse scattering method, darboux transformation method, hirota bilinear method, tanh method and so on. But these methods can't determine the bifurcations of the solutions.In 1966, Peregrine posed the regularized long wave equation (RLW equation in short). This equation is an alternative model to the more familiar Korteweg-de Vries (KDV) equation. Studying this equation is important not only in theory but also in practice.By using the method of planar dynamical systems to a nonlinear variant of the regularized long-wave equation (RLW equation in short), the existence of smooth and non-smooth solitary wave (so called peakon and valleyon) and infinite many periodic wave solutions is shown. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. The formulas to compute the travelling waves are also educed. In the meantime, all of the possible solutions are observed from the qualitative analysis and numerical simulation. Some results in one of the references are incorrect based on the qualitative analysis of this dissertation.