The Adomian Decomposition Method And An Analytic Study Of Explicit Solutions For Some Nonlinear Evolution Equations

In this paper, We have carried on a detailed introduction to the original established thought of Adomian decomposition method and its revision processes, and systematically summarized its application research into nonlinear science and, particularly, the aspect of nonlinear partial differential equation solution.The paper consists of six chapters in total. In chapter one, We have briefly retrospected the historical background and developing processes of soliton, completely summarized the main methods of solving nonlinear partial differential equation brought forward from home and abroad, and also introduced to the point the aim of research and the primary contents of this paper.In chapter two, We have introduced in detail the Adomian decomposition method, the background for its revision, the application operating process of the concrete method, and several kinds of concrete solving methods to the Adomian polynomial and its application.In chapter three, the Adomian decomposition method has been applied to explore the exact solutions to Zakharov-Kuznetsov (ZK(n, n)) equation, obtaining some new periodic travelling wave solutions and compact solutions. We have also carried on its image simulation to the solutions obtained.In chapter four, We have further adopted Adomian decomposition method to obtain the exact and numerical solutions to the revision nonlinear Drinfel'd-Sokolov equation, and also conducted an analysis and research into the exact and numerical solutions obtained, comparing their absolute error to determine precision of the numerical solution obtained.In chapter five, We have adopted the Adomian decomposition method to extract some numerical solutions to long and short wave resonance equation, and carried on a detailed image simulation and numerical analysis to the numerical solutions obtained. At the same time, We have also conducted a research into iterative precision of the decomposition method.In chapter six, We have carried on a detailed analysis and discussion on the Schr(o|¨)dinger-Boussinesq equation bearing Cauchy problem with initial condition and obtained the non-perturbative solution to the equation.Finally, We have provided a summary of this paper and also have looked ahead of research orientation in future.