计算机符号计算在非线性模型解析研究中的应用
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摘要
随着计算机科学与技术的发展,计算机符号计算作为计算机、数学与人工智能的交叉学科逐渐成熟和完善。符号计算研究的主要对象是可代数化、数学化的实际问题,对现实问题进行创造性的数学建模,而对模型的处理需要算法化的数学计算与逻辑推理。通过对实际问题的分析与探讨,设计可以在符号计算软件与系统中实现的代数算法,并分析算法的逻辑性与可执行性,最后应用算法处理模型并得到有效结果。符号计算是非线性模型解析研究的工具,在促进孤子理论的发展方面起了重要的作用。
本文借助符号计算对出现在光纤通信、玻色-爱因斯坦凝聚以及流体力学等领域,具有变系数、耦合性、高维数(2+1或3+1维)以及高阶效应(包括高阶色散作用和高阶非线性作用)的非线性模型做解析研究,并详细探讨相关的可积性质。本文处理非线性模型的主要技术路线有两种：(一)线性化路线：基于可积条件(对变系数模型而言可利用Painleve检测获得系数函数间约束关系),将原非线性模型转化为与之对应的线性系统(或Lax对)的相容性条件,对线性系统构造达布阵(或达布算子),设计纯代数的迭代算法,最后利用符号计算实现想要的结果；(二)双线性化路线：利用Painleve分析或者贝尔多项式处理将非线性模型转化为双线性方程,对双线性方程应用形式参数展开法求解,最后利用符号计算实现想要的结果。在两种技术路线中,对非线性模型相关的可积性质也会做详细探讨。例如,从两种路线出发构造解析多孤子解,基于Lax对构造无穷多个守恒律,通过双线性方程构造双线性形式的Backlund变换以及利用贝尔多项式表达构造贝尔多项式形式的Backlund变换等。
本文的核心内容包括以下七个方面：
(I)借助符号计算,给出了研究非线性模型的双线性方法算法模块。基于双线性方法算法模块双线性化处理了二阶色散变系数非线性Schrodinger (NLS)模型,该模型考虑到光纤中由光纤空间变化引起的非均匀性,可以描述带有分布式参数色散与非线性效应的的单模光纤中脉冲传播的放大与衰减。文中研究了二阶色散变系数NLS模型的解析孤子解以及相关的可积性质,包括双线性Backlund变换,双Wronskian表示,N-1到N孤子之间的迭代等；对光孤子在非均匀光纤中的传播特性、演化行为以及相互作用等动力学特征给出了解析分析和图形模拟。
(Ⅱ)借助符号计算,给出了三个处理非线性模型的算法模块：(a)非线性模型Painleve检测算法模块；(b)构造非线性模型线性系统算法模块；(c)非线性模型Darboux变换算法模块。利用算法模块(a)对广义N耦合高阶效应变系数NLS模型进行Painleve检测得到了变系数函数间两种约束条件：在第一种约束条件下,基于双线性方法算法模块双线性化处理该模型并得到了解析暗孤子解；在第二种约束条件下,首先应用算法模块(b)构造了该模型的线性系统,然后应用算法模块(c)构造了Darboux变换并获得解析亮孤子。通过选取不同的变系数函数,对所得暗／亮孤子作图分析,从理论与图形上揭示非均匀光纤中光孤子的传播演化特点。
(Ⅲ)借助符号计算,对描述单模光纤中非线性脉冲传播的含高阶非线性效应广义NLS模型的各种可积性质(包括调制不稳定性、无穷守恒律、双线性方程以及解析孤子解与分析等)做解析研究。主要研究工作包括：(a)通过给平面波解一个小扰动进行调制不稳定分析；(b)基于其线性系统推导无穷多守恒律；(c)应用符号计算体系下双线性方法算法模块求得解析孤子解；(d)孤子解的渐近分析与图形演化。通过选取双孤子不同的波数,讨论了高阶非线性效应下孤子动力学特征。在相应的参数条件下,运动双孤子的结构从切片图上看类似蚯蚓,基于这一特征将这种双孤子从视觉上形象地命名为蚯蚓子。
(Ⅳ)借助符号计算,基于双线性方法算法模块(并在双线性化过程中引入辅助函数)解析研究两个具有相干耦合效应的NLS类模型,即“负”相干耦合向量孤子模型与新的相干耦合向量孤子模型。对于前者,从向量孤子解的表达式出发进行渐近行为分析,并发现了退化与非退化孤子；在自相位调制、交叉相位调制以及“负”相干耦合等效应的综合作用下,探求向量孤子的形成与传播机制；通过选取不同的相位参数,讨论“负”相干耦合模型退化与非退化孤子间相应于三种碰撞模式下的向量孤子相互作用特点。对于后者,在相应的约束条件下解可以分为两类,即奇异型和非奇异型；非奇异型解具有孤子轮廓。对孤子解的渐近分析与图形模拟刻画了亮向量孤子解的轮廓(单峰或双峰)并揭示其相互作用机制(即在新的相干耦合效应NLS模型中向量孤子仅发生弹性相互作用)。
(V)借助符号计算,解析研究线性的与时相关外部势阱囚禁的准一维玻色-爱因斯坦凝聚体孤子激发态动力学行为。在平均场近似理论下,描述该现象的模型为NLS类方程。利用无量纲变换将该模型无量纲化,并应用Painleve检测算法模块对其进行Painleve检测。基于模型的可积性直接构造精确解析解(包括暗孤子和亮孤子解),并且通过选取系数函数的不同形式对玻色-爱因斯坦凝聚中暗孤子和亮孤子动力学行为进行分情况讨论。
(Ⅵ)借助符号计算,基于Painleve检测算法模块与双线性算法模块解析研究高维孤子问题。涉及的高维(或高维变系数)非线性模型有(2+1)-维Sawada-Kotera模型、带外力项广义二维变系数Korteweg-de Vries (KdV)模型和广义(2+1)-维变系数Gardner模型。对于(2+1)-Sawada-Kotera模型进行Painleve检测并求得解析孤子解,探讨多孤子的传播与相互作用机制。对于带外力项广义二维变系数KdV模型进行双线性研究,获得变系数双线性形式以及双线性Backlund变换；进一步基于双线性Backlund变换的相容性推导了Lax对,得到带外力项广义二维变系数KdV模型在具有Lax对的意义下可积的一些特例。对于广义(2+1)-维变系数Gardner模型,从可积角度出发找到变系数函数间的约束关系,进而对广义Gardner模型进行可积约化；基于可积性对相关的可积性质做了进一步研究,如双线性形式、双线性Backlund变换以及几种不同的Lax对；对应于求解双线性形式、求解双线性Backlund变换以及非线性化Lax对获得了模型的冲击波解。特别地,以二和三冲击波解为例,详尽地分析了冲击波解的传播与相互作用动力学特征,其中着重分析了变系数对冲击波解各个物理量的影响与作用、二冲击波解非弹性作用的机理以及三冲击波解各种非弹性作用(非弹性增益碰撞与非弹性压缩碰撞)的可能情形。
(Ⅶ)借助符号计算,采用经典的贝尔多项式理论处理了如下非线性模型：(1+1)-维浅水波Ⅰ模型、Boiti-Leon-Manna-Pempinelli模型以及(2+1)-维高阶色散Sawada-Kotera模型,并相应地获得了贝尔多项式表达式、贝尔多项式形式的Backlund变换以及Lax对；采用带辅助变量的贝尔多项式方式处理了如下非线性模型：(1+1)-维浅水波Ⅱ模型、Lax五阶KdV模型以及(2+1)-维和(3+1)-维破裂孤子模型,并相应地获得了贝尔多项式形式的Backlund变换。
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.
本文借助符号计算对出现在光纤通信、玻色-爱因斯坦凝聚以及流体力学等领域,具有变系数、耦合性、高维数(2+1或3+1维)以及高阶效应(包括高阶色散作用和高阶非线性作用)的非线性模型做解析研究,并详细探讨相关的可积性质。本文处理非线性模型的主要技术路线有两种：(一)线性化路线：基于可积条件(对变系数模型而言可利用Painleve检测获得系数函数间约束关系),将原非线性模型转化为与之对应的线性系统(或Lax对)的相容性条件,对线性系统构造达布阵(或达布算子),设计纯代数的迭代算法,最后利用符号计算实现想要的结果；(二)双线性化路线：利用Painleve分析或者贝尔多项式处理将非线性模型转化为双线性方程,对双线性方程应用形式参数展开法求解,最后利用符号计算实现想要的结果。在两种技术路线中,对非线性模型相关的可积性质也会做详细探讨。例如,从两种路线出发构造解析多孤子解,基于Lax对构造无穷多个守恒律,通过双线性方程构造双线性形式的Backlund变换以及利用贝尔多项式表达构造贝尔多项式形式的Backlund变换等。
本文的核心内容包括以下七个方面：
(I)借助符号计算,给出了研究非线性模型的双线性方法算法模块。基于双线性方法算法模块双线性化处理了二阶色散变系数非线性Schrodinger (NLS)模型,该模型考虑到光纤中由光纤空间变化引起的非均匀性,可以描述带有分布式参数色散与非线性效应的的单模光纤中脉冲传播的放大与衰减。文中研究了二阶色散变系数NLS模型的解析孤子解以及相关的可积性质,包括双线性Backlund变换,双Wronskian表示,N-1到N孤子之间的迭代等；对光孤子在非均匀光纤中的传播特性、演化行为以及相互作用等动力学特征给出了解析分析和图形模拟。
(Ⅱ)借助符号计算,给出了三个处理非线性模型的算法模块：(a)非线性模型Painleve检测算法模块；(b)构造非线性模型线性系统算法模块；(c)非线性模型Darboux变换算法模块。利用算法模块(a)对广义N耦合高阶效应变系数NLS模型进行Painleve检测得到了变系数函数间两种约束条件：在第一种约束条件下,基于双线性方法算法模块双线性化处理该模型并得到了解析暗孤子解；在第二种约束条件下,首先应用算法模块(b)构造了该模型的线性系统,然后应用算法模块(c)构造了Darboux变换并获得解析亮孤子。通过选取不同的变系数函数,对所得暗／亮孤子作图分析,从理论与图形上揭示非均匀光纤中光孤子的传播演化特点。
(Ⅲ)借助符号计算,对描述单模光纤中非线性脉冲传播的含高阶非线性效应广义NLS模型的各种可积性质(包括调制不稳定性、无穷守恒律、双线性方程以及解析孤子解与分析等)做解析研究。主要研究工作包括：(a)通过给平面波解一个小扰动进行调制不稳定分析；(b)基于其线性系统推导无穷多守恒律；(c)应用符号计算体系下双线性方法算法模块求得解析孤子解；(d)孤子解的渐近分析与图形演化。通过选取双孤子不同的波数,讨论了高阶非线性效应下孤子动力学特征。在相应的参数条件下,运动双孤子的结构从切片图上看类似蚯蚓,基于这一特征将这种双孤子从视觉上形象地命名为蚯蚓子。
(Ⅳ)借助符号计算,基于双线性方法算法模块(并在双线性化过程中引入辅助函数)解析研究两个具有相干耦合效应的NLS类模型,即“负”相干耦合向量孤子模型与新的相干耦合向量孤子模型。对于前者,从向量孤子解的表达式出发进行渐近行为分析,并发现了退化与非退化孤子；在自相位调制、交叉相位调制以及“负”相干耦合等效应的综合作用下,探求向量孤子的形成与传播机制；通过选取不同的相位参数,讨论“负”相干耦合模型退化与非退化孤子间相应于三种碰撞模式下的向量孤子相互作用特点。对于后者,在相应的约束条件下解可以分为两类,即奇异型和非奇异型；非奇异型解具有孤子轮廓。对孤子解的渐近分析与图形模拟刻画了亮向量孤子解的轮廓(单峰或双峰)并揭示其相互作用机制(即在新的相干耦合效应NLS模型中向量孤子仅发生弹性相互作用)。
(V)借助符号计算,解析研究线性的与时相关外部势阱囚禁的准一维玻色-爱因斯坦凝聚体孤子激发态动力学行为。在平均场近似理论下,描述该现象的模型为NLS类方程。利用无量纲变换将该模型无量纲化,并应用Painleve检测算法模块对其进行Painleve检测。基于模型的可积性直接构造精确解析解(包括暗孤子和亮孤子解),并且通过选取系数函数的不同形式对玻色-爱因斯坦凝聚中暗孤子和亮孤子动力学行为进行分情况讨论。
(Ⅵ)借助符号计算,基于Painleve检测算法模块与双线性算法模块解析研究高维孤子问题。涉及的高维(或高维变系数)非线性模型有(2+1)-维Sawada-Kotera模型、带外力项广义二维变系数Korteweg-de Vries (KdV)模型和广义(2+1)-维变系数Gardner模型。对于(2+1)-Sawada-Kotera模型进行Painleve检测并求得解析孤子解,探讨多孤子的传播与相互作用机制。对于带外力项广义二维变系数KdV模型进行双线性研究,获得变系数双线性形式以及双线性Backlund变换；进一步基于双线性Backlund变换的相容性推导了Lax对,得到带外力项广义二维变系数KdV模型在具有Lax对的意义下可积的一些特例。对于广义(2+1)-维变系数Gardner模型,从可积角度出发找到变系数函数间的约束关系,进而对广义Gardner模型进行可积约化；基于可积性对相关的可积性质做了进一步研究,如双线性形式、双线性Backlund变换以及几种不同的Lax对；对应于求解双线性形式、求解双线性Backlund变换以及非线性化Lax对获得了模型的冲击波解。特别地,以二和三冲击波解为例,详尽地分析了冲击波解的传播与相互作用动力学特征,其中着重分析了变系数对冲击波解各个物理量的影响与作用、二冲击波解非弹性作用的机理以及三冲击波解各种非弹性作用(非弹性增益碰撞与非弹性压缩碰撞)的可能情形。
(Ⅶ)借助符号计算,采用经典的贝尔多项式理论处理了如下非线性模型：(1+1)-维浅水波Ⅰ模型、Boiti-Leon-Manna-Pempinelli模型以及(2+1)-维高阶色散Sawada-Kotera模型,并相应地获得了贝尔多项式表达式、贝尔多项式形式的Backlund变换以及Lax对；采用带辅助变量的贝尔多项式方式处理了如下非线性模型：(1+1)-维浅水波Ⅱ模型、Lax五阶KdV模型以及(2+1)-维和(3+1)-维破裂孤子模型,并相应地获得了贝尔多项式形式的Backlund变换。
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.
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