Symmetry And Integrability Of Nonlinear System

Based on symbolic computation, this dissertation investigates the theory with appli-cation of the symmetry and integrability of nonlinear mathematical physics. The main work is carried out in four aspects:the classical symmetry methods are applied to study some important nonlinear models in fluid mechanics as well as the atmosphere and ocean; the theory of nonlocal symmetry is developed and some related applications are real-ized; Bell polynomials are used to analyze the integrability of nonlinear evolution equa-tions with arbitrary parameters; a package of one-dimensional optimal system for finite-dimensional Lie algebra is developed.Chapter1is an introduction to review the theoretical background and development of symmetry theory, integrable system and symbolic computation. The main works of this dissertation are also illustrated.Chapter2concentrates on applying the classical symmetry theory in fluid mechanics as well as atmospheric and oceanic dynamics. Firstly, a novel nonlinear model in fluid mechanics called resonant Davey-Stewartson (DS) system, which is relevant to nonlinear Schrodinger equation, is investigated by some classical symmetry methods. Its continuous and discrete symmetry properties are discussed and three types of reduced equations are obtained. Then, an important nonlinear high-dimensional model in the atmospheric and oceanic dynamics, namely (3+1)-dimensional baroclinic potential vorticity equation is studied. Kinds of exact solutions are obtained from its corresponding lower-dimensional equation. According to these explicit solutions, the Rossby wave with rich vertical struc-ture of the atmospheric motion and some classical radial circulation phenomenons are explained.Chapter3is devoted to developing a new theory of nonlocal symmetry and related applications. Applying the developed nonlocal symmetry method to KdV equation in different forms, many nonlocal symmetries are obtained by taking infinitesimal forms from Backlund transformation, Darboux transformation and bilinear transformation re-spectively. On one hand, abundant explicit solutions of KdV equation are constructed successfully, among which the interactions of elliptic periodic wave and soliton as well as Painleve wave and soliton are firstly discovered. On the other hand, the negative hier-archies of potential KdV equation and other new integrable systems are presented.Chapter4deals with Bell polynomials and related integrabilities. The binary Bell polynomials method is extended to a generalized NNV equation with four arbitrary pa-rameters as well as a nonisospectral and variable-coefficient mKdV equation respectively. Their corresponding Hirota bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws are obtained step by step. This algebraic approach only considers dimensional analysis and elementary combinatorics, without too much clever guesswork.Chapter5focuses on the algorithm of one-dimensional optimal system in symmetry theory. The mechanical algorithm for constructing one-dimensional optimal system of finite-dimensional Lie algebra is firstly provided and corresponding implementation of software package on Maple is accomplished. We summarize and analyze the seemingly random and empirical classification process to design a rule which can be achieved on the computer. Due to our algorithm, a large number of repetitive and tedious calculations are avoided. Lastly, two examples are given to illustrate the effectiveness and practicality of this algorithm.Chapter6concerns the summary and discussion of the whole dissertation, and the prospect for the future work is also put forward.