Study On Constructive Solving Of Some Nonlinear Wave Equations

It is well known that nonlinrar science has become important frontier of science researches.In the recent decades,with the development of the science and technology, all kinds of nonlinear problems lead to interesting for many sciencists and engineers. Some key problems,which appear in modern physics and computation of science engineering,is to find solutions of some specially nonlinear equations indeedly.So solving nonlinear equations occupies an important place in both theory and application. Solving nonlinear equations have been faced with frequent problems for many science researchers.However,constructive solving nonlinear equations is both difficult and important,need effective mathematical tools.In recently years,many researchers do a lot of work to solve nonlinear equations and obtained abundand results.In this paper, using constructive solving principle and given resultswe solve some important nonlinear waves equations,which frequently appears science and engineerings.The paper main contents incluce as fellow:In chapter 1,we introduce the study background,study development and significance of constructive method of nonlinear wave equation.The methods known up to today for solving the nonlinear wave equation are summarized and analyzed.In chapter 2,we study the DBM and Log-DBM equations.Using algebraic methods which include extended Tanh-method,some new solitary solutions to the DBM and Log-DBM equation,such as singular solitary solutions,double singular peakon solitary wave solutions,double solitary wave solution with peakon and singular periodic solitary wave solutions are obtained..With the aid of an auxilitary function combined with the elliptic integral of the first kind,periodic solitary solution,singular and periodic singular solitary solutions can be obtained.In chapter 3,we study the Zakharov-Kuznetsov-Modified Equal-Width (ZK-MEW) equation by employing the extended Jacobi elliptic function expansion method and obtain some new solutions of Jacobi elliptic function type of the ZK-MEW equation.;Secondly,using sn-cn method,we study K(m,n,1) equation and obtained some new exact solutions when k=s=3,especially obtained abundand solitary solutions and trigonometric function solutions when m→0 and m→1;In chapter 4,Painleve singular analysis was applied to nonlinear PDE with variable coefficients.As an example,Painleve singular analysis is discussed for Burger equation with variable coefficients and damping,obtained the Backlund transformation and some new soltary solutions.The homogenous balance(HB) method is used to obtain the Backlund transformation,and some new solitary solutions of Log-DBM,include peakon solutions,are given by the transformation.Using a new transformation,the compaeton-like and solitary patterns-like solutions of Log-DBM equation are obtained.In chapter 5,Lie group analysis method was applied to nonlinear PDE with variable coefficients.Using this method,we discussed for generalized Burger equation with variable coefficients and linear danping,obtained its symmetry group,simliar reduced equations,infinitesimal generator and Lie algebras;In chapter 6,using the Exp-function method,the generalized solutions of combined KdV-mKdV equation with variable coefficients are discussed.The many exact solutions and compact-like solutions uch as,trigonometric function,solitary pattern solutions, solitary wave solutions are given by the auxiliary equation method.By compared with other method,this method has the merits of short computation and abundant results, which can be applied to solve other equations with variable coefficients.In chapter 7,the summary and expectation are given.