Symbolic Computation On The Integrable Properties And Physical Applications Of Some Nonlinear Models

As a new branch of artificial intelligence, computerized symbolic computation has been an effective assistant tool for the study of nonlinear science. On symbolic system, computerized symbolic computation can exactly and effectively perform inference operation in terms of algorithms, which finally leads to the mechanized solving of research problems. With the rapid development of computer technology and symbolic computation systems, the investigations on the integrable properties and physical applications of nonlinear models have become to be one important research direction in nonlinear science.Based on symbolic computation, this dissertation is to investigate the integrable properties and physical applications of some nonlinear models from certain physical setting. Most of nonlinear models investigated here are the multi-component, higher-dimensional and nonintegrable equations which are difficult to be solved and appear of wide application in various branches of physics and engineering technology. With the aid of symbolic computation, this dissertation is to algorithmically construct the solutions and investigate integrable properties of some nonlinear models enjoying the features involved above. With the proposal and construction of some algorithms of concerning problems, certain typical examples are carried out on symbolic system. This dissertation focuses on the proposal and implementation of some algorithms, and particularly emphasizes on the analysis and discussion of some nonlinear phenomena and mechanisms described by nonlinear models in certain physical setting. Furthermore, we can explain the propagation characteristics and motion laws of the solitons occurring in many fields of physical and engineering sciences through the analytic analysis.The main body of this dissertation is based on the 12 SCI-journal papers of the first author published or accepted during the period for the author to pursue the Ph.D. degree work. The research work of this dissertation was supported by the Excellent Ph.D. Students Innovation Fund of Beijing University of Posts and Telecommunications (No. CX200902). The detailed research work mainly includes the following aspects:(1) Based on computerized symbolic computation, the Ablowitz-Kaup-Newell-Segur (AKNS) system is generalized to the multi-block matrix form. Thus, to some extent, this form extends the application area of AKNS system. One not only can derive the linear eigenvalue problem associated with the scalar nonlinear models, but also can construct the linear system of multi-component nonlinear models. Thereby, the integrable properties of multi-component models can be investigated, and their analytic multi-soliton solutions are also constructed on the basis of the linear system. In illustration, the Lax pairs associated with the multi-component nonlinear Schrodinger and modified Korteweg-de Vries equations are derived in the form of block matrices.(2) Symbolic computation on the Darboux iterative algorithm for constructing the solutions of multi-component nonlinear models has been proposed. When the Darboux transformation method is applied to the Lax pair associated with the multi-component systems, the key point encountered is how to make the reduction and constraints among original potentials invariable. As far as we know, although there is not a systematic and effective method to deal with this problem, the Darboux transformation of some integrable multi-component systems can be constructed according to the characteristics and symmetry space of their linear systems. With the help of symbolic computation, the Darboux iterative algorithm is applied to the N-component modified Korteweg-de Vries model and N-component nonlinear Schrodinger model. The N-soliton solutions of multi-component nonlinear models are derived in terms of Vandermonde-like determinant by iteratively solving algorithm equation on symbolic computation system. Moreover, the formulas of the N-soliton solution in Vandermonde-like determinant dramatically reduce the complexity of computational operation.(3) For the nonlinear models in the same hierarchy, they have series of important and common characteristics. Therefore, it is interesting and meaningful to investigate the nonlinear hierarchies, not limited to a single nonlinear model. Based on the (2+1)-dimensional AKNS system, the (2+1)-dimensional nonlinear Schrodinger hierarchy is derived and the Darboux iterative algorithm is applied to the Lax pair associated with this hierarchy. The general explicit formula for N-soliton solution is derived in terms of Vandermonde-like determinant expression. In addition, the interactions between the solitons are analyzed including line-line, parabola-parabola and line-parabola types.(4) The singular manifold method is an important approach to investigate the integrable properties for many integrable nonlinear models in soliton theory. In this dissertation, the two-singular manifold method is applied to the (2+1)-dimensional Gardner equation with the achievement of some integrable properties, e.g., the Hirota bilinear form, Backlund transformation, Lax pairs and Darboux transformation. Based on the obtained Lax pairs, the binary Darboux transformation is constructed and the N×N Grammian solution is also derived by performing the iterative algorithm N times with symbolic computation.(5) Symbolic computation study on the Landau-Lifshitz-type nonlinear model describing the nonlinear dynamics of a Heisenberg spin chain with an external time-oscillating magnetic field. For the nonlinear dynamics produced by the interaction between the electronmagnetic wave and magnetic medium in magnetic spin chain, of particular concern are the propagation characteristics, production mechanism and application prospect. Under the influence of an external time-dependent magnetic field, the propagation characteristics, motion dynamics and particle property for solitons become the focus of attention. With the help of symbolic computation, the analytic multi-soliton solutions are obtained by applying the Darboux iterative algorithm to the corresponding linear spectral problem of Landau-Lifshitz-type nonlinear model. By virtue of obtained soliton solutions, the effects of an external magnetic field on the solitons are discussed.(6) With the aid of symbolic computation, the bilinear method is extended to investigate some complicated multi-component nonlinear models such as the coupled nonlinear Schrodinger equations governing the propagation of vector solitons in nonlinear optical fibers. According to the obtained multi-vector soliton solutions, the interactions between vector solitons are discussed by making the asymptotic analysis and calculating some important physical quantities. The collisions with complete or partial switching of energy between vector solitons are investigated in multi-component nonlinear models. Accordingly, the possible physical applications of collision dynamics between vector solitons are pointed out such as the operation of optical controlled optical nonlinear logical gates, the design of fiber directional couplers and quantum information processors.(7) Based on the symbolic computation, the bilinear method is applied to the higher-dimensional or coupled nonintegrable nonlinear models. Generally speaking, in nonlinear science, it is difficult to obtain the analytic soliton solutions of these higher-dimensional or coupled nonintegrable nonlinear models. With the combination of the features of bilinear method and symbolic computation, the nonintegrable (2+1)-dimensional Boussinesq and (2+1)-dimensional coupled Schrodinger models are investigated in gravity water wave and nonlinear optical fiber, respectively. The analytic one-and two-soliton solutions of such two nonintegrable nonlinear models are constructed, and abundant (2+1)-dimensional collision behaviors are investigated.(8) Symbolic-computation study on the transmission characteristics and interactions of ultrashort soliton pulses in inhomogeneous fibers. With the aid of symbolic computation, the bilinear method is applied to the nonlinear Schrodinger equation with varying coefficients. The propagation and interactions of ultrashort soliton pulses are discussed based on the obtained analytic solution solutions. The results could provide some theoretical bases for studying the stable transmission of soliton pulses in inhomogeneous fibers or dispersion-managed fiber transmission systems.