Invariant Subspace Of Nonlinear Partial Differential Equations And Their Exact Solutions

In the early eighteenth and nineteenth century, one of the crucial problems in the theory of partial differential equations (PDEs) was finding and studying the classification of important equations that were integrable in closed form and, in particular, possessed explicit solutions. Therefore, solving these equations is particularly important. One of the solving methods is the separation of variables, which was developed by Fourier in the study of heat conduction problems, and was later generalized and extended by Sturm and Liouville.The invariant subspace method is also called generalized variable separation and it is an universal method of solving partial differential equation. Many interesting exac-t solutions of nonlinear partial differential equations in various fields can be achieved through the invariant subspace method. The reason why this method gets more attention is based on the fact that high-dimensional nonlinear partial differential equations typically have variable separation solutions. The key advantage of the invariant subspace method for solving nonlinear partial differential equations is that we can construct subspaces of partial differential equations using the solutions of linear ordinary differential equations. Its main idea is to convert an initial partial differential equation to the ordinary differen-tial operator. This thesis is to introduce the general steps for solving nonlinear evolution equation by invariant subspace method.The symbolic computation system plays an important role in getting exact solution by solving nonlinear partial differential equation. This thesis will simplify the solving process armed with symbolic computation software Maple. We will give the3D graphs of the exact solutions by Maple. It enables us to study the meaning, properties of the solutions more easily.In this thesis, we mainly study from the following several aspects:In section1, we introduce the background and significance of the research.Section2is the basis of this thesis. First of all, we should understand two most fundamental and most important problems of the invariant subspace field. Secondly, we introduce the basic classification of invariant subspace and then give a summarization.Section3mainly introduces three aspects:1. The dimension of the subspace and the largest dimension;2. Summarize the operators which admit the polynomial subspace; 3. The operator of sub-maximal order and translation invariant operator are dis-cussed.Section4introduces the steps of solving differential equations with variable coef-ficients by invariant subspace. In terms of the invariant subspace method and under the assumption that the dimension of invariant subspace is two, a number of two-dimensional invariant subspaces are obtained and a series of different separation of variable solutions are given correspondingly. At the same time, the symbolic solving algorithm is given based on the symbolic computation system Maple such that the method can be easily extended from low dimension to high dimension cases.Section5is about the expansion of the invariant subspace, and the subspace which depends on the independent variable t and the local invariant subspace are studied.Section6is a summary of this thesis and the prospect of solving nonlinear evolution equations.