Application Of Computerized Symbolic Computation To Analytic Study On Some Nonlinear Models

With the development of computer science and technology, computerized symbolic computation, as an interdisciplinary subject of computer science, mathematics and artificial intelligence, has gradually become ripe and perfect. The main research objects of symbolic computation are the algebraization and mathematization of practical questions, which involves the creative construction of mathematical modeling, and the manipulation on the models with algorithmization of mathematical calculation and logical reasoning. Through the analysis and exploration of practical problems, algebra algorithm can be designed and implemented on symbolic computation softwares and systems; moreover, the logicality and performability of the algebra algorithm should be analyzed, and finally valid results on the models will be obtained with the use of the algebra algorithm. Symbolic computation is the tool of the analytic study on the nonlinear models, and plays an important role in the development of soliton theory.With symbolic computation, this dissertation is to analytically investigate certain variable-coefficient, coupled, higher-dimensional [(2+1)-and (3+1)-dimensional] and/or higher-order effect (higher-order dispersion and higher-order nonlinearity) involved nonlinear models, which appear in the optical communication, Bose-Einstein condensates and fluid dynamics. Furthermore, relevant integrable properties are also studied in detail. The main technical routes of this dissertation dealing with the nonlinear models are of two types:(A) Linearization route:Cast the original nonlinear models as the compatibility conditions of their corresponding linear systems (or Lax pair) based on the integrable constraints (which can be derived with Painleve test for the variable-coefficient models), construct Darboux matrix (or Darboux operator), design the purely algebraic iterative algorithm, and finally achieve the expected results with symbolic computation;(B) Bilinearization route:Transform the nonlinear models into bilinear equations with Painleve analysis and/or Bell-polynomial manipulation, solve the bilinear equations with the formal parameter expansion method, and finally achieve the expected results with symbolic computation. In these two types of technical routes, the relevant integrable properties of the nonlinear models are also discussed detailedly, such as solving the analytic multisoliton solutions with these two types of technical routes, deriving the infinite conservation laws based on the Lax pair, constructing the bilinear Backlund transformation via bilinear equations, and obtaining the Bell-polynomial-typed Backlund transformations from Bell-polynomial manipulation.The research work of this dissertation includes the following seven aspects:(I) In virtue of symbolic computation, the bilinear method algorithm module is given to investigate the nonlinear models. Based on this algorithm module, a generalized variable-coefficient nonlinear Schrodinger (NLS) model is bilinearized, which considers the heterogeneity originated from the spacial changes in optical fibers and can describe the amplification or attenuation of pulse propagation in a single-mode optical fiber with distributed dispersion and nonlinearity. This dissertation investigates the analytic soliton solutions and associated integrable properties including bilinear Backlund transformation, double Wronskian expression, and transformation from the (N-1)-to N-soliton solutions. Additionally, the dynamical properties of the optical soliton propagation, evolution and interaction behavior in inhomogeneous fiber are analytically discussed and graphically simulated. (II) In virtue of symbolic computation, three algorithm modules for dealing with nonlinear models are given:(a) Painleve test algorithm module for nonlinear models;(b) linear system construction algorithm module for nonlinear models;(c) Darboux transformation algorithm module for nonlinear models. Painleve test is carried out for the generalized variable-coefficient N-coupled higher order NLS system via the algorithm module (a) with the achievement of two types of constraints among the variable-coefficient functions:Under the first type of constraints, the generalized variable-coefficient N-coupled higher order NLS system is bilinearized via the bilinear method algorithm module with the achievement of analytic dark-soliton solutions; under the second type of constraints, the associated linear system is firstly constructed via the algorithm module (b), and then the bright-soliton solutions are derived by means of Darboux transformation via the algorithm module (c). With the different choices among the variable-coefficient functions, the dark-and bright-soliton solutions are graphically analyzed, and the propagation and evolution of the optical solitons in the inhomogeneous optical fibers are theoretically and graphically revealed.(III) In virtue of symbolic computation, this dissertation analytically investigates the associated integrable properties (modulation instability analysis, infinite conservation laws, bilinear equation and analytic soliton solution, etc.) of a generalized higher-order nonlinear effect involved NLS model, which can be used to describe the propagation of nonlinear pulse in a monomode optical fiber. The main research work focuses on (a) modulation instability analysis of solutions in the presence of a small perturbation;(b) derivation of the infinite conservation laws based on the Lax pair;(c) soliton solutions obtained in virtue of the bilinear method with symbolic computation;(d) asymptotic analysis and graphical illustration of the solitons. Solitonic characteristics are discussed with different choices of the wave numbers in the two-soliton solutions. Finally a new type of soliton, namely the "earthwormon" is proposed in that the moving two-soliton structure looks like an earthworm in slice graphics.(IV) In virtue of symbolic computation, this dissertation analytically investigates two coherently-coupled NLS-typed models, i.e., a NLS-typed model with negative coherent coupling and a new coherently-coupled NLS-typed model. By means of the bilinear method algorithm module with the introduction of an auxiliary function, analytic vector bright solitons are derived. For the former model, asymptotic behavior analysis are carried out based on the expressions of the vector two-soliton solutions, and the degenerate/non-degenerate vector solitons are found; with the combined effects of self-phase modulation, cross-phase modulation and negative coherent coupling, the formation and transmission mechanisms of the vector solitons are studied; with different choices of the phase parameters, the interaction features of coherently-coupled degenerate and non-degenerate vector solitons, associated with three types of interaction models, are discussed. For the second model, its solutions are classified under corresponding constraints as two types:singular and non-singular ones, and the later ones appear as soliton-typed; asymptotic behavior analysis and graphical simulation for the solitons indicate the profiles of the bright vector solitons (single-or double-hump ones) and reveal their interaction mechanisms (that is, only elastic interactions take on in vector solitons of the new coherently-coupled NLS-typed model).(V) In virtue of symbolic computation, this dissertation analytically investigates the dynamics of soliton excitations in quasi-one-dimensional Bose-Einstein condensates trapped with an arbitrary linear time-dependent potential. In the mean field approximation theory, this phenomenon is described with NLS-typed equation. With a dimensionless transformation the model is cast into a dimensionless one, and is analyzed via Painleve test algorithm module. Within its integrability, exact analytic solutions including dark-and bright-soliton ones are directly constructed, and their dynamic behaviors in Bose-Einstein condensates are discussed with different choices of the arbitrary linear time-dependent potential.(VI) In virtue of symbolic computation, this dissertation analytically investigates higher-dimensional soliton problems via Painlev e test and bilinear method algorithm modules. The concerned higher-dimensional and variable-coefficient nonlinear models are:the (2+1)-dimensional Sawada-Kotera model, generalized variable-coefficient two-dimensional Korteweg-de Vries (KdV) model with various external-force terms, and generalized (2+1)-dimensional variable-coefficient Gardner model. For the (2+1)-dimensional Sawada-Kotera model, Painlev e test is carried out, analytic soliton solutions are solved, and the soliton propagation and interaction are revealed as well. The bilinear research is conducted for the generalized variable-coefficient two-dimensional KdV model with various external-force terms, and as a result, the bilinear form and bilinear Backlund transformation are derived. Furthermore, Lax pairs are constructed with the compatibility of the bilinear Backlund transformation, and some Lax-integrable cases of the generalized variable-coefficient two-dimensional KdV model with various external-force terms are obtained. For the generalized (2+1)-dimensional variable-coefficient Gardner model, the constraints among the variable-coefficient functions are found in the viewpoint of integrability, under which the generalized model is reduced. Other related integrable properties such as bilinear form, bilinear Backlund transformation and several different Lax representations are further studied. Through solving bilinear equations, bilinear Backlund transformation and nonlinearization of the Lax pair, shock-wave-like solutions are given. Specially, dynamics of the propagation and interaction for two-and three-shock-wave-like solutions are detailedly analyzed as an example, among which, the key points are the analysis of the influence of variable-coefficient functions on the physical quantities, the inelastic interaction mechanisms of the two-shock-wave-like solutions, and all the possible interaction cases (inelastic amplification and/or inelastic compression interactions) for the three-shock-wave-like solutions.(VII) In virtue of symbolic computation, this dissertation on one hand applies the classic Bell-polynomial theory to three nonlinear models:(1+1)-dimensional shallow water wave model, Boiti-Leon-Manna-Pempinelli model and the (2+1)-dimensional Sawada-Kotera model with the achievement of their corresponding Bell-polynomial expressions, Bell-polynomial-typed Backlund transformations and Lax pairs. On the other hand, by means of the Bell-polynomial manipulation with an auxiliary independent variable, the Bell-polynomial-typed Backlund transformations for another (1+1)-dimensional shallow water wave model, Lax fifth-order KdV model,(2+1)-and (3+1)-dimensional breaking soliton models are constructed as well.