Invariant Subspaces And Exact Solutions Of Three Types Nonlinear Evolution Equations

Considered herein is the exact solutions of nonlinear partial di?erential equations by using invariant subspace method. In case of the continuous equations, the invariant subspace admitted by the equations is firstly found and then the exact solutions of the equations are figured out by invariant subspace method. While in case of discrete equations, the basic idea is introduced under the condition that the time-variables is continuous and the spatial-variables is discrete.The corresponding lower dimensional reduction of the finite-di?erence solutions on the invariant subspace are constructed. We obtain the exact solution that will enhance to learn the properties of the equations. In this paper, two types of invariant subspace for the polynomial subspace and trigonometric subspace are discussed in detail, as an example to the KZK, LRT and m ZK equations. Finally, the di?erence for the exact solutions of these two types of invariant in continuous and discrete cases for the nonlinear partial di?erential equations.An outline of this thesis is presented as follows:Chapter 1 is a preliminary chapter, which encompasses the basic theory of invariant subspace method for the continuous and discrete equations.In Chapter 2, we mainly discuss the exact solution for a class of nonlinear evolution equations. By using invariant subspace method for the continuous and discrete equations, we observe the exact solutions of KZK equation in these two cases.In Chapter 3, it shows the LRT equation of invariant subspace with polynomial subspace,and in the continuous and discrete equations, the exact solutions of LRT equation is obtained.In Chapter 4, considered herein is the m ZK equation of invariant subspace with polynomial subspace. In both the continuous and discrete equations, the exact solutions of m ZK equation is turned out in detail.In Chapter 5, it makes a summary and discusses what can be done in the future.