Symmetries And Solitary Wave Solutions And Similarity Solutions Of Evolution Equations

We often meet a lot of nonlinear evolution equations in mathematics and physics.Finding their solutions are much a difficult and a challenge question. Many scientificworkers did a lot preeminent contribution for this. The Lie group analyzing method and thefunctional transformation method are classical methods, which are a theoretical andpractical significance in studying of differential equations.By the Lie group analyzing method, we may study invariance nature of thenonlinear evolution equations. We reduce equation using the invariance nature and find itsinvariance solutions. The key is its infinitesimal generators i.e. its symmetries(by solvingthe set of determining equations or more the characteristic set for the infinitesimalgenerators admitted by nonlinear evolution equations). Therefore symmetries play a greatimportant role in solving partial differential equations.The functional transformation method changes the original equation into easysolving equation by proper transformation. Tanh—function method or generalizdtanh—function method is a quite efficient method.In this thesis, (1) using Chao lu's procedure packages of determining thedifferential Characteristic set theory of differential polynomial system, we obtain thedifferential Characteristic set of determining equations for symmetry and finally determinetheir symmetries of 26 partial differential equations; (2) we use generalized thetanh-method, i.e. the sechq-tanhq method, and give solitary wave solutions of theN-component nonlinear Schrodinger, Klein-Gordon and Kdv equation in the followingforms, (u, uv~2,...,uv~r,...,uv~N), where u=sechq(), v=tanhq(). For the N-component nonlinearSchrodinger and Klein-Gordon equations, we give a type of solitary wave solutions whenN is odd ; (3) we get a new type of solitary wave solutions of the variable coefficient Kdvequation.