A Study Of Construction Methods For Analytical Solutions To Some Nonlinear Differential Equations

In this dissertation,two kinds of problems are considered,one is about the improvement of construction methods for analytical solutions to nonlinear differential equations,the other is about the applications of these methods in specific problems.This work is divided into five Chapters:In Chapter 1,history and current progress of soliton theory,nonlinear bubbles and con-struction methods for analytical solutions are introduced,from which the main work of the dissertation is presented.Chapter 2 is devoted to the improvement of the homotopy analysis method.Two kind of modified techniques are discussed:on one hand,to overcome the problem that the traditional homotopy analysis solution expressed by power series is often effective only in a local region,theory of formal power series is employed to obtain an unified solution formula on each subin-terval,and finally we construct a piecewise homotopy analysis solution which is effective in a more large region;on the other hand,by introducing the idea of Newton iteration,the obtained optimal homotopy analysis solution is applied to revise the initial guess,which significantly increases the convergence rate and accuracy of the series solution.In Chapter 3,with the aid of the Hirota bilinear equations,quasi-periodic wave solutions for the nonlocal Boussinesq equation are constructed,and also,the relations between the quasi-periodic wave solutions and the corresponding soliton solutions are given by a limiting proce-dure.Chapter 4 deals with the construction of analytical solutions for the Rayleigh-Plesset equa-tion,which describes the bubble dynamics in an incompressible fluid.By means of different construction methods,many kinds of analytical solutions are given,based on which the law of bubble motion are also discussed.Chapter 5 summarizes the main results of the dissertation,and prospects the research prob-lems in future.