Analysis Of Solitary Waves For A Class Of Shallow Water Wave Systems

Shallow water wave systems can theoretically grasp the law of tides, swells and the dam collapses in Harbors, water pollution, tsunami etc.. With the needs of ocean development, disaster prevention, environmental protection, the study of the shallow water wave systems has become a hot topic. The study on properties of solitary waves is of great value in scientific researchers and applications to explain the wave propagation and natural phenomena as well as to determine the physical attributes of materials.A class of shallow water wave systems are studies in this dissertation. This system consists of a series of shallow water wave equations which have close relationship with KdV equation. The research of this dissertation includes Painleve property of systems, Backlund transformation, traveling waves classification, the stability of solitary waves and related properties.The first chapter is devoted to the background and current situation.of shallow water wave systems and the research of tsunamis models.In chapter 2, we study the Painleve property of shallow water wave systems. By using WTC-Kruskal method, a type of systems with Painleve property are obtained. Furthermore, by considering the influence of nonlinear term coefficients to Painleve property, two new types of systems with Painleve property are derived.In chapter 3, the construction of explicit solitary waves of shallow water wave systems is investigated. We improved some classical methods to find solutions of shallow water wave systems with complex nonlinear terms, and we obtain abundant solitary wave solutions, especially we obtain abundant new solitary wave solutions: compacton solutions, kink-compacton solutions, non exponential peakon solutions and composed solutions.In chapter 4, the Backlund transformation of shallow water wave systems is discussed. Motivated by an algebraic method seeking solutions, we give a new construction of the Backlund transformation between solutions of two equations. By using improved homogeneous balance method, the Backlund transformation between two solutions of the same equation. is studied. Some new solitary solutions are obtained, such as double solitary wave solutions with peakon or singularity. By painleve truncation method, the auto-Backlund transformation is given and the transformation is consistent with the result by using the improved homogeneous balance method.In chapter 5, the classification of the traveling wave solutions of shallow water wave systems is studied. By using qualitative analysis of extreme points and singularity points, we classify traveling wave solutions to two kinds of shallow water wave systems with marvelous structure. Abundant solutions are determined, among which some are solitary solutions with special forms. Improvements in classification analysis lie in that the influence of convection term coefficients to the forms and distribution of solutions is considered, the condition of smooth solutions converge to non-smooth solutions is studied, and new types of explicit solutions are obtained by the qualitative analysis method.In chapter 6, the orbital stability and linear stability of the solitary wave solutions of shallow water wave systems is discussed. Some important conservalation laws and Hamilton structure are given. The orbital stability of smooth solitary solutions and linear stability of compacton solutions are studied, and the solutions are stable under some certain conditions.In chapter 7, the application of shallow water wave systems in the research of tsunamis models is studied. By summarizing the scientific research achievement, the relationship of tsunamis and shallow water wave, the generation mechanism of tsunamis, the basic characteristics of tsunami waves and the research progress of tsunami source models and tsunami propagation models are introduced. The research is hoped to offer some theoretical insights for the further research on tsunami models.Chapter 8 is a summary and expectation.