The Research On The Solution And The Bifurcation Of A Class Of Nonlinear Partial Differential Equations

Nonlinear dynamics, which mainly studying the bifurcation,chaos, fractal, solit-onand so on,is an important branch of the nonlinear subjects.The exact solution and theresearch about the solution of the nonlinear partial differential equations are the maincontent of the nonlinear dynamics.and At present, some achievements of the exactsolutions and the method of solving, which having become a cutting-edge research innonlinear science topics and hot issues,of nonlinear partial differential equationshave been obtained.The theory and the algorithms of the exact solutions have beenproposed and developed. And it’s very important and valuable to look for the effectivesolution.In this paper, the solving methods of nonlinear partial differential equationshave been studied systematicly and deeply. And a exact solution,solving the physicsand mechanics of nonlinear partial differential equations,is introduced.The existresults and some new exact solutions are also obtained. Therefore, the work of thispaper is theoreticaly significant and applicational valuable.There are five chapters in this paper.The first chapter is an introduction, includi-ng the solving method of the exact solution of the nonlinear equations and thestudying perpose,the main content and innovation. The second chapter describes anew method for solving nonlinear partial differential equations that is auxiliarydifferential equation method. First, the basic principle of auxiliary differential methodwas introduced and then the algorithm of auxiliary differential method was described.The solitary wave solutions, periodic solutions and doubly periodic solutions of theequation can be obtained by this method. The third chapter introduces the (2+1)-dimension dispersive long wave equation, and then its exact solution can beobtained by the auxiliary differential method. The exact solutions, including solitarywave solutions and periodic solutions, were analyzed by the Mathematic software.The graphics, drawn by the software, of the solitary wave solutions and periodicsolutions were helpful to the analysis of the solution. The (2+1)-dimensionKonopelchenko-Dubrovsky bifurcation equation was obtained in the fifth chapter.And the bifurcation analysis of the bifurcation equation was done. The bifurcationresponse curses which the system having saddle-node bifurcation can be got fromwere drowned. There are multiple values of frequency response, jumping and otherdynamic behavior. The influence of the excitation amplitude to the bifurcation response curves. Finally, this article summarizes the work and future researchdirections are also discussed.