Nonlinear wave equations and solitary waves

The goal of this dissertation is to study nonlinear wave theories, in particular solitary wave theory. Some new results of the RLW equation were carried out and discussed in Chapters II-III, which can help understand the different properties between the KdV (Korteweg-de Vries) equation and the RLW (Regularised Long Wave) equation. A new explanation for the collisions of the KdV solitary waves and the RLW solitary waves was provided in Chapter IV.; A simple transformation which converts nonlinear waves (both bell solitary wave and kink solitary wave) into sinusoidal (linear) waves was found (Chapter V). A detailed discussion about that transformation and its application to polynomial nonlinear evolution equations was given in Chapter VI. New solutions to some well-known equations were found (Chapter VII).; Finally, two new decompositions for the KdV two-solitary wave solution were discussed in Chapter VIII.