Oscillation And The Exact Solutions Of Nonlinear Differential Equations

The theory of oscillation is an important branch of the qualitative the-ory of diferential equations,and it has a profound efect in the establishmentand research of mathematical model,the research and application of physical the-ory and engineering theory.Since S.Hilger[1] proposed the theory of measurementchain,the theory of dynamic equations on time scales is getting a lot of atten-tion and having wide development.In recent years,the study of oscillation theoryhave got rapid development.Diferential equations have expanded high order dif-ferential equations!functional diferential equations and dynamic equations ontime scales.Many scholars pay more attentions to it, have achieved many goodresults(see [1]-[24]).Exact solutions of Nonlinear Evolution Equations get a lot of mathemati-cians and physicists’ attention,they are valuable on the theory and applicationresearch.In recent years,as the symbol of computer software development,peopleput forward many kinds of efective methods to get exact solutions of Non-linear Evolution Equations,Such as Backlund Transform!Painleve expansionmethod!tanh function method!The homogeneous balance method!Auxiliaryequation method[25-40],thus promote the rapid development of Nonlinear Evolu-tion Equations.The present paper employed a generalized Riccati transformation,the ba-sic Inequality method and Integral average method for some class of diferentialequations, we got some new oscillation criteria. In addition,with the help of newsolutions of the auxiliary equation,we got some new rational form solutions ofNonlinear Evolution Equations.This thesis is divided into three sections.In Chapter1,Preface, we introduce the main contents of this paper.In Chapter2section1,we study the oscillation for second order nonlineardamped diferential equation of the formx (t)+p(t)x (t)+f(t, x[τ1(t)],..., x[τm(t)], x [δ1(t)],..., x [δm(t)])=0, t≥t0we mainly employed generalized Riccati transformation,Inequality method andIntegral average technique,introduced constant factor β>1to zoom up and down,improved the main results of others, got some new oscillation criteria and took an example to explain what we obtained.In Chapter2section2,we studied the oscillation for second order nonlinear damped dynamic differential equation on time scales of the form: we employed generalized Riccati transformation,average function technique and Integral inequality method,improved the main results of [19,22,23],and obtained the new oscillation criteria.In Chapter3section1,we used the solutions of auxiliary equation and a generalized transformation to solve the variant Boussiesq equation we got a series of new exact solutions,this method can be applied to other Nonlinear Evolution Equations.In Chapter3section2,we presented a new extended Jacobi elliptic function expansion(EJEFE) method to solve the (2+1)-dimensional Davey-Stewartson equation and successfully obtained some new exact doubly periodic solutions in terms of rational form,and avoided the complicated calculation of Jacobi elliptic function.