Application Of Symbolic Computation To Some Nonlinear Mathematical Models

Based on symbolic computation and the knowledge of differential equations, algebra and operators, this dissertation presents an interdisciplinary study of some important nonlinear mathematical models, arising from such applications as optical soliton communications, transfers of energy in biophysics, space plasmas and Bose-Einstein condensates.The nonlinear science is one of the most popular fields of current scientific researches, and its branches include soliton, chaos and fractal. The main objective of this dissertation focuses on higher-order, couple and variable coefficient nonlinear mathematical models which are applicable to the description of various soliton phenomena. By virtue of the powerful symbolic computation technology, an analytic investigation is performed on these models from the aspects of bilinear representation, Backlund transformation, Wronskian solution and infinite conservation laws.The results obtained in this dissertation are as follows:( I ) Based on the basic principles of the bilinear method, via symbolic computation, we deduce some formulas which can conveniently transform original nonlinear differential equations into the corresponding bilinear ones. Then using these transformation formulas, we respectively transform three different types of nonlinear Schrodinger equations, i.e., nonisospectral, higher-order and couple higher-order ones, into the corresponding bilinear forms. From these bilinear equations, we can get multi-soliton solutions by employing the expansion method of small parameter. Furthermore, these bilinear equations also provide a basis for deriving bilinear Backlund transformations. (II) Based on the basic idea of Backlund transformation, via symbolic computation, we derive a series of the bilinear exchange formulas which are used to deduce Backlund transformation. It is mentioned that the derivation of the exchange formulas is necessary but very difficult for constructing the bilinear Backlund transformation. Using a suitable exchange formula, we successfully derive Backlund transformations for three different types of nonlinear Schrodinger equations, i.e., nonisospectral, higher-order and couple higher-order ones. Via symbolic computation and starting from a trivial solution, the Backlund transformation yields the analytical soliton solution of interest in physics. As an example, we detailedly illustrate and discuss the soliton features of a nonisospectral nonlinear Schrodinger equation. On the other hand, based on the Backlund transformation, the author also derives the inverse scattering transform scheme for a higher-order nonlinear Schrodinger equation.(III) Based on the definition and theory of Wronskian determinant, via symbolic computation, we present the Wronskian solutions of three different types of nonlinear Schrodinger equations, i.e., nonisospectral, higher-order and couple higher-order ones, and verify them by direct substitution into the bilinear equations. On the other hand, the authors also verify that the (N-1)- and N-soliton solutions satisfy the Backlund transformation with sets of parametric conditions. In the study of solitons, the multi-soliton collision is an important issue. The soliton in Wronskian form can directly give explicit multi-soliton solutions, so it becomes easier to study the dynamics of the multi-soliton collision. As an example, we give a detailed analysis on characteristics of the two-soliton solution of a nonisospectral nonlinear Schrodinger equation.(IV) With the aid of symbolic computation, the author proposes a generalized Miura transformation which relates the solutions of the variable-coefficient Korteweg-de Vries equation to those of a variable-coefficient modified Korteweg-de Vries equation. Then by using such a Miura transformation and the Galilean invariant transformation, the author proves the existence of infinite conservation laws under the Painleve integrable condition. The nonlinear mathematical models investigated in this dissertation have a wide range of applications in modern science and technology. The author hopes that the results, obtained by symbolic computation and the theory of differential equations, algebra, operators and other relevant mathematical theory, have some contribution to the study of nonlinear science. It is expected that our work can provide the theoretical guidance in the real world for optical soliton communications, transfers of energy in biophysics, space plasmas, Bose-Einstein condensates, etc., and well be of value in the development of differential equations and symbolic computation technique.