Symmetry Study And Its Applications Of Several Nonlinear Equations

In this dissertation, symmetry group study and some methods concerned with symmetry group for some nonlinear differential equations in fluid dynamics are investigated through the symmetry theory.Chapter 1 is a brief review of the background and development of symmetry theory and group theory with explanations of some concepts related to symmetry group whose generalizations or combinations form the foundation for the whole dissertation.In Chapter 2, by using the Clarkson and Kruskal's direct method, all the pos-sibilities of similarity reductions are obtained for the coupled KdV equations and the corresponding explanations of group theory are provided. Using Ablowitz-Ramani-Segur (ARS) algorithm, the coupled KdV systems are reclassified under the Painleve integrable sense.Chapter 3 is mainly focused on the one-dimensional nonlinear Schrodinger equation with a perturbation of polynomial type. Using the approximate symme-try perturbation theory, the approximate symmetries and approximate symmetry reduction equations are obtained. But because of the complex of the reduction equation, we failed to derive the series reduction solution of the perturbed NLS equation.Chapter 4 discusses the conservation laws of the KP equation which are close related to symmetry. We construct conservation laws of the equation family which possesses same infinite dimensional Kac-Moody-Virasoro symmetry alge-bra as the Kadomtsev-Petviashvili (KP) equation. The conservation laws are calculated up to second order group invariants and described by two arbitrary functions of six variables and one arbitrary function with four variables. In Chapter 5, the symmetry transformation group of the bilinear negative Kadomtsev-Petviashvili system is studied by means of the modified CK's direct method. By taking the limitation of the transformation group, the Kac-Moody-Virasoro type Lie point symmetry algebra is found which is proved to be a special infinitesimal form of the symmetry group.Chapter 6 introduces a simple method of getting the symmetry group and symmetry algebra of a nonlinear system by using of Lax pair. (2+1)-dimensional Euler equation is considered. A new symmetry group theorem of the equation is obtained which is thus utilized to find exact analytical vortex and circumfluence solutions. A weak Darboux transformation theorem of the (2+1)-dimensional EE can be obtained for arbitrary spectral parameter from the general symmetry group theorem. The possible applications of the vortex and circumfluence solu-tions to the tropical cyclones, especially the Hurricane Katrina 2005 and Neoguri 2008, are demonstrated.The last chapter concerns the summary and discussion for the whole disser-tation, as well as the prospect for those methods.