Lie Symmetry And Invariant Subspace Method To Some Nonlinear Evolution Equations

In this dissertation, Lie symmetry analysis is applied to study the high-order nonlinear thin film equation in fluid mechanics. The invariant groups, optimal systems and some physical interest solutions of the equation are de-rived. In addition, by applying the invariant subspace method, we inves-tigate the most general third-order nonlinear differential operators and the two-component nonlinear cross-diffusion systems with convection and source terms, classification theorems of them which possess different invariant sub-spaces are also constructed. The main results are as follows:1. We utilize the classical Lie symmetry to discuss the 2m-order nonlinear thin film equation. The corresponding infinitesimal generators, the optimal systems and the group-invariant solutions are obtained. Then we analyze some physical interest solutions.2. By the invariant subspace method, we consider the most general third-order nonlinear differential operators F(x,u,ux,uxx,uxxx). The classification consists of three steps. First, it is shown that they are quadratic forms while preserving invariant subspace of maximal dimension. The full descriptions of the third-order quadratic operators with constant coefficients which pos-sess maximal dimensional invariant subspaces are given. Then after a similar analysis, we discuss the classification of the third-order nonlinear differen-tial operators F(x, u, ux, uxx, uxxx) with subspaces of submaximal dimension. At last, classification theorems of the third-order nonlinear quadratic oper- ators which preserve three-, four-, five-dimensional invariant subspaces are constructed.3. By extending invariant subspace method, we investigate the classifi-cation of nonlinear cross-diffusion systems with convection and source terms The nonlinear vector differential operators (F1,F2) preserve invariant sub-spaces Wn11×Wn22 governed by the system of ordinary differential equations where 0<n≤n1,n1≤9, n1-n2≤2. A large list of systems together with their invariant subspaces Wn11×Wn22 is provided.