Symmetry Reductions And Exact Solutions Of Several Nonlinear Problems

Based on the classical Lie group theories and computer symbolic computation, this dissertation investigates the optimal system of group invariant solutions and the corre-sponding algorithms, realizing part of this algorithm procedure on the symbolic com-putation system-Maple; Combining the classical Lie group method and the generalized direct method successfully, we investigates the problem of solving some important non-linear evolution equations exactly and the analysis of the graph simulation obtained by computer. For example, we study some atmospheric dynamics equations including the Navier-Stocks equations; find some exact solutions which can reflect some important nat-ural phenomena such as a double-eyewall typhoon solution.Chapter 1 introduces the background and the development of the symmetry theories, the optimal system of group invariant solutions, the symbolic computation and the models of oceanic and atmospheric dynamics. A brief introduction of the results achieved by the domestic and foreign scholars in these areas is given at last.Chapter 2 makes use of the classical Lie group method and its algorithms, com-bining with the generalized direct method to investigate the (1+1)-dimensional hyper-bolic Monge-Ampere equation, the (3+1)-dimensional Zakharov-Kuznetsov equation, the (2+1)-dimensional dispersive long wave equations and the generalized Nizhnik-Novikov-Veselov equations. An Maple's arithmetic of the Killing form is given, which is used in finding the optimal system of group invariant solutions. We investigate the one-, two-, and three-parameter optimal system of group invariant solutions and give out the correspond-ing reductions of the one-, two-, and three-dimensional subalgebras. The relationship between one solution and another is obtained by the generalized direct method, further to get new solutions of the original equation. The graph analysis of the new exact so-lutions obtained here is given out. Meanwhile, the conservation laws are studied of the (2+1)-dimensional dispersive long wave equations.Chapter 3 studies three equations of the ocean and atmosphere:the Navier-Stokes equations, the two layer model equations and the two layer model equations with external forcing heating function, of which the last two are presented the first time by us. By using the optimal system of group invariant solutions and its algorithms, one-dimensional or two-dimensional subalgebras of these three equations are classified and the corresponding reductions and solutions are given. For some interesting explicit solutions, the figures are given out to show their properties. The typhoon solution is obtained successfully. And we find the picture that can be used to simulate a double-eyewall structure of typhoon firstly. Chapter 4 gives out the symmetry reductions of a discrete equation:the Blaszak-Marciniak four-field lattice equation by means of the classical Lie group method. And the solutions which have not only the one-soliton structure but also the singularity are obtained. For the mKdV equation and the modified Zakharov-Kuznetsov equation with the initial value problem, the homotopy perturbation method is used to get the analytical approximate solutions. Choosing the form of the initial value, the single soliton, two-soliton and rational solutions are presented. Applying the classical Lie group method into the two initial value problems, the solutions are obtained, which are analyzed and compared with that obtained by using the homotopy perturbation method.The last chapter concerns the summary and discussion for the whole work of this dis-sertation, including the advantages and disadvantages of the work, as well as the prospect for the next work.