Symbolic Computation On The Integrable Properties Of Some Variable-Coefficient Nonlinear Models

With the swift development of the computerized symbolic computation, the analytic investigation on variable-coefficient nonlinear models in nonlinear sciences has become one important research direction in soliton theory, especially for the integrable property issue. Due to the practicability of symbolic computation, it can drastically increase the ability of a researcher to algorithmically deal with the complicated and interminable analytic calculations, and then provide a wonderful tool for the study of variable-coefficient nonlinear evolution equations (NLEEs).Based on symbolic computation, the main work of this dissertation is to generalize some known methods that are used to treat constant-coefficient NLEEs to study the integrable properties of some variable-coefficient nonlinear models, including the variable-coefficient Korteweg-de Vries (KdV) model, variable-coefficient higher-order nonlinear Schrodinger (NLS) model, variable-coefficient modified KdV-Sine-Gordon (mKdV-SG) model, variable-coefficient Gardner model, cylindrical Kadomtsev-Petviashvili (KP) model and nonisospectral modified KP (mKP) model. Those models with variable coefficients have wide application prospects in various branches of physics and engineering technology such as optical fiber communications, plasmas, superconductors, hydrodynamics and nonlinear lattice, especially for describing the physical mechanism of nonlinear phenomena in physical situations with nonuniformities of boundaries and/or inhomogeneities of media.In this dissertation, the author emphasizes the proposal of some algorithms, which are convenient to be operated on computing system and can be used to explore the integrable properties of some variable-coefficient nonlinear models. Moreover, with the use of integrated software for mathematical computing, special attention is paid to some obtained analytic results as well as their physical mechanism and possible applications. The research work of the dissertation mainly includes the following aspects:(1) Variable-coefficient Ablowitz-Kaup-Newell-Segur (AKNS) method with symbolic computation. Owing to the limitation of the constant-coefficient AKNS method, a variable-coefficient AKNS algorithm is proposed for constructing the Lax pairs of variable-coefficient nonlinear models, which admits exact, algorithmic and exercisable traits and is applicable to large numbers of variable-coefficient NLEEs. In illustration, the algorithm is applied to some variable-coefficient nonlinear models arising from plasma physics, nonlinear lattice, Bose-Einstein condensates and ocean dynamics. Utilizing this effective variable-coefficient algorithm, one can directly investigate the integrable properties of variable-coefficient NLEEs, and simultaneously obtain the Lax pairs of both the constant-coefficient and variable-coefficient models.(2) Symbolic computation on the auto-Backlund transformations and multi-soliton-like solutions of variable-coefficient nonlinear models. From the viewpoint of the equivalent relationship between the auto-Backlund transformation and the inverse scattering system, a systematic algorithm is put forward to construct the corresponding auto-Backlund transformations and one-soliton-like solutions for some variable-coefficient NLEEs. Meanwhile, the nonlinear superposition formula, two-soliton-like solution and an infinite number of conservation laws of a generalized variable-coefficient KdV equation arising from arterial mechanism and Bose-Einstein condensates are also obtained. On the other hand, under certain constraint conditions, several transformations from the variable-coefficient KdV equation and the variable-coefficient Gardner equation to their constant-coefficient integrable counterparts are respectively derived with the help of symbolic computation. By virtue of the obtained transformations, some integrable properties of these two variable-coefficient models are investigated, such as the auto-Backlund transformations, nonlinear superposition formulas, soliton-like solutions and Lax pairs. Via Mathematica software, the explicit physical properties and latent applications of the obtained analytic solutions are graphically discussed.(3) Darboux transformations for the variable-coefficient nonlinear models with symbolic computation. Compared with the classical Backlund transformation, the Darboux transformation possesses an obvious advantage that there simultaneously exist the potential function transformation and wave function transformation. Thereby, one can derive a series of new analytic solutions of NLEEs from an initial solution in the purely algebraic manner via the computerized symbolic computation. By combining some characteristics of the symbolic computation, Darboux transformation and variable-coefficient integrable models, an algorithm is proposed and applied to construct the Darboux transformations for these models. Consequently, the Nth-iterated potential transformation formula and multi-soliton solutions are also obtained. Especially, with symbolic computation, the double singular manifold method is extended to the variable-coefficient nonisospectral mKP equation aiming to derive the auto-Backlund transformation, a couple of Lax pairs, binary Darboux transformation and the Nth-iterated potential transformation formula in the form of Grammian for such a model.(4) Symbolic computation on the integrable decompositions for the cylindrical KP equation. Via the decomposition method and symbolic computation, the author is devoted to working on the cylindrical KP model for investigating the significant parabola dust-acoustic wave soliton structures occurring in the dusty plasmas and Bose-Einstein condensates. Firstly, through considering suitable symmetry constraints between the Lax pair and adjoint Lax pair of the cylindrical KP model, two kinds of integrable decompositions are proposed, i.e., the nonlinearization of a single Lax pair and the nonlinearization of two symmetry Lax pairs. Secondly, another integrable decomposition is directly presented by taking into account the relationship between the cylindrical KP model and the (l+l)-dimensional integrable soliton systems. Through these three kinds of integrable decompositions, such a variable-coefficient (2+1)-dimensional model can be respectively decomposed into two variable-coefficient (1+1)-dimensional integrable soliton systems, which are the first two non-trivial equations of the same hierarchy. Therefore, the solutions for the higher dimensional complicated systems can be reduced to solve the lower dimensional simple integrable systems. This provides us with a way to investigate the properties of the former based on the latter. Based on the Lax representations for these (1+1)-dimensional integrable systems, several Darboux transformations are constructed to iteratively generate a rich class of analytic soliton-like solutions. In virtue of the powerful plot function of Mathematica software, the relevant physical mechanisms and possible applications are explicitly discussed through the figures for some sample solutions, which suggests that abundant and interesting dust-acoustic wave soliton structures can occur in the dusty plasmas and Bose-Einstein condensates, such as the one-parabola soliton structure, compressive and rarefactive oblique soliton resonance phenomena, and the dust-acoustic wave soliton elastic interactions.(5) Symbolic computation on the integrable properties for a generalized variable-coefficient higher order NLS model, which has important and wide applications in nonlinear optical fiber systems. Two kinds of general constraint conditions are symbolically derived by employing the Painleve analysis for this model to possess the soliton solutions. It is also found that one of these two constraint conditions is consistent with the result presented in the existing literature, under which some integrable properties have been widely investigated. Thus, the author focuses on studying some remarkable properties for the variable-coefficient higher order NLS model under another set of constraints, e.g., the3×3 matrix Lax pair, Darboux transformation and multi-soliton-like solutions. Furthermore, through controlling the corresponding physical parameters like the self-steepening and amplification/absorption effects in the femtosecond soliton control systems, some features of femtosecond solitons and potential applications in the inhomogeneous optical fiber systems are graphically discussed by the one-and two-soliton-like solutions.In conclusion, with the aid of symbolic computation, the particular processes of several algorithms suitable for investigating the integrable properties for some variable-coefficient nonlinear models are explicitly presented. Accordingly, the possible physical applications of the obtained analytic results are also graphically discussed. It is hoped that these algorithms presented in this dissertation, such as the variable-coefficient AKNS method, algorithm for deriving the atuo-Backlund transformations, methods for constructing Darboux transformations and multi-soliton-like solutions for variable-coefficient nonlinear models, might be valuable and helpful for investigating other variable-coefficient NLEEs. Meanwhile, it is also expected that the analytic results and relevant discussions on the soliton-like solutions in this dissertation will be observed in the future space and laboratory experiments, and can be used to explain some physical mechanisms of various phenomena occurring in optical fiber communications, superconductors, nonlinear lattice, hydrodynamics, dusty plasmas and Bose-Einstein condensates.