【作者】 李娟；

【导师】 田播；

【作者基本信息】 北京邮电大学 ， 计算机软件与理论， 2008， 博士

【摘要】 随着计算机符号计算的迅猛发展,在非线性科学中,基于符号计算的变系数模型的解析研究已逐渐成为孤子理论的重要研究方向之一,特别是关于变系数模型可积性质的研究备受关注。计算机符号计算具有易于操作和实现的特点,能够以算法化的形式处理繁复冗长的表达式,可为变系数非线性发展方程的研究工作提供功能强大的辅助工具。本文主要借助计算机符号计算将某些适用于常系数非线性发展方程的方法进行推广,并应用于若干变系数模型,解析地研究它们的可积性质,如变系数Korteweg-de Vries(KdV)模型、变系数高阶非线性Schr(?)dinger(NLS)模型、修正KdV-Sine-Gordon(mKdV-SG)模型、变系数Gardner模型、柱Kadomtsev-Petviashvili(KP)模型和谱可变修正KP(mKP)模型等等。这些变系数模型在物理学和工程技术领域的不同分支中都有着广泛的应用,如光纤通信、等离子体、超导体、流体力学和非线性晶格等,特别是可用于描述带有非均匀边界条件或非均匀介质的物理背景中的各种非线性波动现象的动力学机制。在本文中,作者主要针对变系数可积模型的研究提出便于计算机符号计算实现的算法,并将其应用于若干变系数模型可积性质的研究之中。此外,借助数学计算软件着重探讨所得解析结果的物理机制和潜在的物理应用。本文的内容主要包括如下几个方面:(一)基于符号计算的变系数Ablowitz-Kaup-Newell-Segur(AKNS)方法:本文对常系数AKNS方法进行适当推广,并结合符号计算提出适用于构建变系数可积系统Lax对的算法,这样,可在很大程度上扩展该方法的适用范围。同时以在等离子体物理、非线性晶格、玻色爱因斯坦凝聚和海洋动力机制等领域中有着广泛应用的变系数模型为例,阐述了变系数AKNS算法的有效性和适用性。利用该变系数算法,可以更直接更有效地研究变系数非线性模型的某些性质,不仅可以获得常系数模型的线性系统,而且还可以构建变系数可积模型的Lax对。(二)基于计算机符号计算的变系数非线性模型自-B(?)cklund变换和多孤子型解的研究:一方面,利用自-B(?)cklund变换与逆散射方法之间的等价关系,提出由变系数可积模型的Lax对推导自-B(?)cklund变换的系统算法,实现对变系数模型可积性质的直接有效的分析探讨,并获得相应的自-B(?)cklund变换和单孤子型解;另一方面,在一定的约束条件下,借助符号计算分别推出从变系数KdV模型和变系数Gardner模型到与它们相对应的常系数可积模型的坐标变换,进而基于此研究了变系数非线性发展方程的孤子解及某些可积性质,如自-B(?)cklund变换、非线性叠加公式和Lax对等。文中还以一类源于动脉机制和玻色爱因斯坦凝聚等物理领域的广义的变系数KdV模型为例,借助符号计算推出该模型的非线性叠加公式和无穷守恒律,并得到双孤波型解。借助Mathematica软件对解析解的潜在物理应用进行直观地分析讨论。(三)符号计算与变系数非线性模型的Darboux变换:与经典B(?)cklund变换相比,Darboux变换的一个显著优势就是既含有势函数变换又存在波函数变换,可反复利用同一个算法迭代推出非线性发展方程的一系列解析解。文中主要结合符号计算、Darboux变换、变系数非线性模型的某些特点,提出构造变系数可积模型Darboux变换的算法,并得到相应模型N次迭代的势函数变换公式及多孤子型解。特别地,将双奇异流形方法推广应用到具有双Painlevé分支的变系数谱可变mKP模型,借助符号计算得到该可积模型的自-B(?)cklund变换、Lax对、Darboux变换和Grammian形式的解析解。(四)基于符号计算的柱KP模型的Darboux变换和多孤子型解的研究:主要利用非线性化方法和符号计算研究柱KP模型所描述的尘埃等离子体和玻色爱因斯坦凝聚中的抛物线型尘埃声波孤子结构。首先,从柱KP模型的Lax对及其共轭Lax对着手,构造两者之间合适的对称约束并提出两种可积分解,即单个Lax对的非线性化和两个对称Lax对的非线性化;其次,从考察柱KP模型与(1+1)维可积方程之间关系的角度直接构造可积分解。这三种可积分解分别将柱KP模型分解为同一梯队中的两组(1+1)维变系数可积系统,从而可降低模型的维数,达到利用低维方程研究高维复杂方程的目的。借助符号计算获得分解后的(1+1)维可积系统的几种Darboux变换,进而得到柱KP模型一系列的孤子型解。借助Mathematica软件分析讨论了单抛物线型孤子、稀疏型和压缩型孤子的共振结构及离子声波孤子碰撞结构在尘埃等离子体和玻色爱因斯坦凝聚中的物理机制和潜在的物理应用。(五)利用计算机符号计算重点研究在非线性光纤光学中有着重要应用的变系数高阶NLS模型的可积性质:通过Painlevé分析方法得到该模型存在脉冲孤子解的两种系数约束条件,在其中一种条件下,该模型的一些可积性质已被广泛研究。本文着重探讨变系数高阶NLS模型在另外一种约束条件下具有的可积性质,包括3×3矩阵Lax对、Darboux变换和多孤子型解。通过对解析解中物理参数(如自陡峭效应和光纤增益/损耗效应)的合理取值,并借助单孤子型解和双孤子型解的几组图形,直观详细地讨论了飞秒光孤子脉冲在非均匀光纤系统中的某些特性及其潜在应用。综上所述,本论文针对变系数模型含有任意变系数函数的特点,基于符号计算提出了若干研究变系数模型可积性质的算法,并利用Mathematica计算软件对所得解析结果的潜在物理应用进行了深入分析。作者希望文中提出的用于研究变系数模型可积性质的方法,如变系数AKNS方法、推导自-B(?)cklund变换的方法、构造Darboux变换的方法及得到变系数模型多孤子型解的方法,能够为其它类型的变系数非线性模型的研究工作提供一定的帮助。同时也希望本文获得的解析结果及关于孤子型解的分析讨论,有可能在未来的空间和实验室环境中被观察到,并有助于解释光纤通信、超导体、非线性晶格、流体力学、尘埃等离子体和玻色爱因斯坦凝聚等领域中非线性现象的物理机制。更多还原

【Abstract】 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.更多还原