Study On The Decomposition And Interact Of Soliton Solutions For Some Nonlinear Evolution Equations

Solitons exist widely on many nonlinear phenomena and they play an important role in development of modern science and technology. Soliton theory—a theory of nonlinear partial differential equations (nonlinear PDEs)—started to evolve more than forty years ago. Today this theory is a hot researching topic and mathematically beautiful, but still developing fast, driven by many applications in various fields of physical science and engineering. Soliton phenomena exist on two conditions. First, there must exist stable solitary waves that travel with constant configurations (shape, speed, etc.) as long as they do not meet any external obstacles. Second, if a solitary wave meets another of its kind, they interact, but without destroying each other's identities (elastic interaction). Such solitary waves are called solitons. The soliton phenomenon is essentially a nonlinear phenomenon. The aim of this thesis is to find a detailed description of soliton solutions of some nonlinear evolution equations by using the Hirota bilinear method, decomposition of multi-soliton solutions and their interactions, in order to make the best use of them.Generally, in applied mathematics, we can understand that solitons are local travelling wave solutions of nonlinear evolution equations and their wave shape and speed do not change but phase may shift after they collide mutually. But in physics domain, solitons can be understood that they are solitary wave which shape and speed change faintly, or they are energy limited solutions of nonlinear evolution equations, namely energy centers in the limited area of space and don't spread to the infinity of space as time goes.Applications of soliton solutions is extremely widespread, in particular their applications receive the widespread research in nonlinear fiber optical solitons communication in the recent years. Therefore, it is especially essential that we do research on soliton solutions, decomposition and interactions of them. It has the actual application value, too.This thesis studies one kind of solutions——soliton solutions of nonlinear evolution equations. It is based on the theory of nonlinear evolution equations, combines with fast developing soliton theory and with the help of mathematic software. Mainly completed work are as follow: firstly, we analyze and summarize types of solitons and introduce multi-soliton and infinite conservational laws of solitons. We can study better solitons and their qualities through above-mentioned elementary knowledge. Secondly, we study soliton solutions of nonlinear evolution equations by using the Hirota bilinear method. We give the Boussinesq equation and the (3+1) dimensional KP equation as examples to discuss the application of the Hirota method which is used to solve soliton solutions of nonlinear evolution equations. At the same time, we also give a simplified Hirota method and its application. Finally, we study a decomposable method of multi-soliton solutions and their interactions and give two cases of soliton interactions by figures. We also compare two different interactional processes. We hope that wecan understand further soliton solutions of nonlinear evolution equations, qualities of solitons and their applications.