Nonlocal Symmetry, Exact Solutions And Integrable Properties Of Several(2+1)-Dimensional Nonlinear Equations

In nonlinear mathematical physics, nonlinear equations are important mathematical models for describing complex physical phenomena in various ?elds of sciences. In this dissertation, by means of computer algebra, some problems of nonlinear equations are studied, including nonlocal symmetry, exact solutions and integrability. The main work is carried out as follows:Chapter 1 is an introduction to review the research background and development of the soliton theory. Some works and achievements that have been obtained are presented. The main works of this dissertation are also illustrated.Chapter 2 considers the nonlocal symmetry and exact solutions of nonlinear equation.The nonlocal symmetry of the(2+1)-dimensional breaking soliton equation is derived from the Lax pair. Under a variable transformation, this nonlocal symmetry becomes residual symmetry which is obtained by the truncated Painlev′e analysis. The residual symmetry is readily localized to Lie point symmetry by prolonging the original equation to a larger system. Meanwhile, the interaction solutions between the soliton and the cnoidal periodic wave are investigated.Chapter 3 is devoted to ?nding the soliton-cnoidal wave solutions of the(2+1)-dimensional breaking soliton equation by using the consistent tanh expansion method. Despite the simplicity of the consistent tanh expansion method, it did provide us with the result which is quite nontrivial and di?cult to be obtained by other traditional approaches.Chapter 4 concentrates on investigating the integrability of a(2+1)-dimensional variablecoe?cient CDGKS equation through Bell polynomials. The bilinear form, bilinear B¨acklund transformation, Lax pair and in?nite conservation laws of this equation are systematically obtained. The integrable constraint conditions on variable coe?cients can be naturally obtained in the procedure of applying the Bell polynomials approach. Moreover, the N-soliton solutions of the equation are constructed with the help of the Hirota bilinear method.Chapter 5 is the summary and outlook of this dissertation.