非线性系统的可积性分析及孤子的相互作用研究
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摘要
孤子是非线性科学的一个重要分支,在数学、物理等领域有广泛应用,并且在金融领域可作为研究市场演化特征的理论基础。因此,孤子理论的研究具有重要意义。
本文主要解析研究非线性系统的可积性及孤子的相互作用机制。通过对变系数、耦合非线性发展方程及方程族的研究,获得一系列研究结果,如非线性系统的可积性、解析孤子解、Hamilton结构及Liouville可积性等。本文共分为六个方面：
(1) Darboux变换的构造及其在耦合非线性发展方程中的应用。(a)分别以Hirota-Maxwell-Bloch (H-MB)方程及广义非均匀H-MB方程、非均匀耦合非线性Schrodinger(NLS)方程为例,构造等谱和非等谱可积系统的Darboux变换；(b)利用Darboux变换构造H-MB方程的单孤子和双孤子解,由此研究孤子的产生机制、传播特性及相互作用；绘图分析广义非均匀H-MB方程的非均匀因素对孤子的发展特性及相互作用的影响；通过控制群速度色散、自相位调制、交叉相位调制及增益／损耗对应的参数,讨论非均匀耦合NLS方程孤子的相互作用机制及在非均匀光纤系统中的潜在应用；(c)利用Painleve检测确定广义非均匀H-MB方程的可积条件；基于Ablowitz-Kaup-Newell-Segur (AKNS)系统,构造H-MB方程及广义非均匀H-MB方程的Lax对；将2×2等谱AKNS谱问题推广到3×3非等谱情形,求得非均匀耦合NLS方程的Lax对。
(2)广义非均匀H-MB方程的N次Darboux变换构造及孤子解的渐近分析。(a)构造广义非均匀H-MB方程的N次Darboux变换,得到方程的单孤子、双孤子及三孤子解,并整理为行列式形式；(b)讨论取不同参数值时孤子的传播特性及相互作用,并发现双孤子碰撞存在能量交换的现象；(c)利用渐近分析研究孤子碰撞前后的物理量,如能量、振幅、脉冲宽度、传播速度和初始相位；(d)给出该方程的前三个守恒律。
(3)非线性发展方程族与无穷守恒律。(a)以KdV和AKNS方程族为例,介绍Lax对为算子和矩阵形式所对应的非线性发展方程族的构造过程；(b)以KdV和AKNS系统为例,研究Lax对为算子和矩阵形式所对应的无穷守恒律的构造过程,并推出H-MB方程的无穷守恒律；(c)以推广的离散谱问题为基础,推导离散系统的方程族及无穷守恒律。
(4) Jaulent-Miodek(JM)谱问题的Hamilton结构、Darboux变换及新的类孤子解。(a)基于JM谱问题,推出JM方程族,并构造该JM方程族两种形式的Darboux变换；(b)利用辛—逆辛分解法,得到JM方程族的Hamilton结构,并证明该方程族在Liouville意义下是完全可积的；(c)通过两种形式的Darboux变换,可以分别构造JM方程族新的类孤子解,这些类孤子都由冲击波和钟形孤子构成；(d)研究两种类孤子的传播特性及相互作用。
(5)等谱和一阶非等谱Kaup-Newell(KN)方程族之间的规范变换。(a)简要介绍规范变换的基本概念,引入等谱和一阶非等谱AKNS系统,得到等谱和一阶非等谱KN谱问题之间的规范变换；(b)求得等谱和一阶非等谱KN方程族之间的转化关系,并给出两个方程族的前三个方程；(c)以等谱和一阶非等谱KN系统为例,研究同一系统对应的等谱和非等谱谱问题之间的直接规范变换。
(6)双线性方法及多线性分离变量法在(2+1)维色散长水波方程中的应用。(a)介绍双线性化常用的三种因变量变换及其它形式的因变量变换,以及相应的非线性发展方程的类型；(b)利用多线性分离变量法研究非线性局域激发模式,并列举几种常见局域解的函数形式；(c)对(2+1)维色散长水波方程进行Painleve分析,可得Painleve展开存在单奇异流形和双奇异流形展开两种形式,进而得到两种不同形式的因变量变换；(d)利用两种因变量变换,分别将方程双线性化和线性化,进而求得方程的解析孤子解及非线性局域激发模式,并通过绘图探讨孤子的传播特性和相互作用。
Soliton is an important branch of nonlinear science. It is widely used in mathematics and physics, and is also the theoretical foundation in financial field for studying the characteristics of market evolution. Therefore, the study of soliton theory is of great significance.
This dissertation analytically investigates the integrability and soliton interaction mechanism in nonlinear systems. Through the analysis on some coupled and variable-coefficient nonlinear evolution equations (NLEEs) or nonlinear hierarchies, a series of results are derived, such as the integrable properties, analytical soliton solutions, Hamil-tonian structure and Liouville integrability. This dissertation mainly includes the fol-lowing six parts:
(1) The construction of Darboux transformation (DT) and its applications in the cou-pled NLEEs.(a) The construction of DT for isospectral and nonisospectral integrable systems is investigated, taking the Hirota-Maxwell-Bloch (H-MB) equation, generalized inhomogeneous H-MB equation and inhomogeneous coupled NLS equation for example;(b) Based on the obtained one-and two-soliton solutions of the H-MB equation via the DT, the production mechanism, propagation characteristics and soliton interactions are investigated. The analysis is made to probe the influence of inhomogeneities in the generalized inhomogeneous H-MB equation on the soliton propagation and interac-tions via some figures. With the corresponding parameters of group velocity dispersion, self-phase modulation, cross-phase modulation and gain/loss under control, the soliton interaction mechanism and its potential applications of the inhomogeneous coupled NLS equations are discussed;(c) Via the Painleve analysis, the integrable conditions for the generalized inhomogeneous H-MB equation are obtained. Based on the Ablowitz-Kaup-Newell-Segur (AKNS) system, the Lax pairs associated with the H-MB equations and generalized inhomogeneous H-MB equations are constructed. With the2×2isospectral AKNS spectral problem generalized to the3×3nonisospectral case, the Lax pair for the inhomogeneous coupled NLS equations is derived.
(2) The construction of N-fold DT and asymptotic analysis of the generalized in-homogeneous H-MB equation.(a) The N-fold DT for the generalized inhomogeneous H-MB equation is derived, as well as one-, two-and three-soliton solutions, which can be compiled into the determinant forms;(b) Soliton propagation characteristics and in-teractions with different parameters are discussed, and energy redistribution between the two solitons in their interactions is observed;(c) The asymptotic analysis is used to discuss some physical quantities before and after soliton interactions, such as the energy, amplitude, width, velocity and initial phase;(d) The first three conservation laws of the equation are obtained.
(3) Nonlinear hierarchies and infinite conservation laws.(a) By taking KdV and AKNS hierarchies for example, the processes of constructing the nonlinear hierarchies with the Lax pair in operator or matrix forms are introduced;(b) By taking KdV and AKNS systems for example, the construction processes of the infinite conservation laws with the Lax pair in operator or matrix forms are investigated, and the infinite conservation laws for H-MB equation are also obtained;(c) Based on the generalized discrete spectral problem, the nonlinear hierarchy and infinite conservation laws of the discrete system are derived.
(4) Hamiltonian structure, Darboux transformation and new soliton-like solutions of the Jaulent-Miodek (JM) hierarchy;(a) Based on the JM spectral problem, the integrable JM hierarchy is obtained, and two types of DTs of the hierarchy are constructed;(b) Via the symplectic-cosymplectic factorization, the Hamiltonian structure of JM hierarchy is obtained, and the hierarchy is proved to be completely integrable in the Liouville sense;(c) Through the two type of DTs, new soliton-like solutions are derived, which are all composed of the shock wave and bell-shaped soliton;(d) Propagation characteristics and interactions of the obtained solitons are graphically discussed.
(5) Gauge transformation between the isospectral and first-order nonisospectral Kaup-Newell (KN) hierarchies.(a) The basic concept of gauge transformation is briefly intro-duced. The gauge transformation between the isospectral and first-order nonisospectral KN spectral problems is derived by introducing the isospectral and first-order non-isospectral AKNS spectral problems;(b) The equivalence between the isospectral and first-order nonisospectral KN hierarchies, as well as the first three representative equa-tions, is given;(c) By taking the isospectral and first-order nonisospectral KN systems for example, the direct gauge transformation between the corresponding isospectral and nonisospectral problems of the same system is investigated.
(6) The applications of the bilinear method and multi-linear variable separation ap-proach in (2+1) dimensional dispersive long wave (DLW) equation.(a) Three types of the dependent variable transformations used in the bilinearization, as well as other de-pendent variable transformations, are introduced, and the corresponding types of NLEEs are also presented;(b) The multi-linear variable separation approach is used to investi-gate nonlinear localized excitations, and several common functions for localized solutions are briefly introduced;(c) The Painleve analysis is performed on the (2+1) dimensional DLW equation, and two types of expansion, i.e., one and two singular manifolds, are found. Correspondingly, two different forms of the dependent variable transformation are derived;(d) Via the two types of the dependent variable transformations, the DLW equation is respectively bilinearized and linearized, and further the analytic soliton solu-tions and nonlinear localized excitations are obtained correspondingly. Besides, figures are made to probe the soliton propagation characteristics and interactions.
本文主要解析研究非线性系统的可积性及孤子的相互作用机制。通过对变系数、耦合非线性发展方程及方程族的研究,获得一系列研究结果,如非线性系统的可积性、解析孤子解、Hamilton结构及Liouville可积性等。本文共分为六个方面：
(1) Darboux变换的构造及其在耦合非线性发展方程中的应用。(a)分别以Hirota-Maxwell-Bloch (H-MB)方程及广义非均匀H-MB方程、非均匀耦合非线性Schrodinger(NLS)方程为例,构造等谱和非等谱可积系统的Darboux变换；(b)利用Darboux变换构造H-MB方程的单孤子和双孤子解,由此研究孤子的产生机制、传播特性及相互作用；绘图分析广义非均匀H-MB方程的非均匀因素对孤子的发展特性及相互作用的影响；通过控制群速度色散、自相位调制、交叉相位调制及增益／损耗对应的参数,讨论非均匀耦合NLS方程孤子的相互作用机制及在非均匀光纤系统中的潜在应用；(c)利用Painleve检测确定广义非均匀H-MB方程的可积条件；基于Ablowitz-Kaup-Newell-Segur (AKNS)系统,构造H-MB方程及广义非均匀H-MB方程的Lax对；将2×2等谱AKNS谱问题推广到3×3非等谱情形,求得非均匀耦合NLS方程的Lax对。
(2)广义非均匀H-MB方程的N次Darboux变换构造及孤子解的渐近分析。(a)构造广义非均匀H-MB方程的N次Darboux变换,得到方程的单孤子、双孤子及三孤子解,并整理为行列式形式；(b)讨论取不同参数值时孤子的传播特性及相互作用,并发现双孤子碰撞存在能量交换的现象；(c)利用渐近分析研究孤子碰撞前后的物理量,如能量、振幅、脉冲宽度、传播速度和初始相位；(d)给出该方程的前三个守恒律。
(3)非线性发展方程族与无穷守恒律。(a)以KdV和AKNS方程族为例,介绍Lax对为算子和矩阵形式所对应的非线性发展方程族的构造过程；(b)以KdV和AKNS系统为例,研究Lax对为算子和矩阵形式所对应的无穷守恒律的构造过程,并推出H-MB方程的无穷守恒律；(c)以推广的离散谱问题为基础,推导离散系统的方程族及无穷守恒律。
(4) Jaulent-Miodek(JM)谱问题的Hamilton结构、Darboux变换及新的类孤子解。(a)基于JM谱问题,推出JM方程族,并构造该JM方程族两种形式的Darboux变换；(b)利用辛—逆辛分解法,得到JM方程族的Hamilton结构,并证明该方程族在Liouville意义下是完全可积的；(c)通过两种形式的Darboux变换,可以分别构造JM方程族新的类孤子解,这些类孤子都由冲击波和钟形孤子构成；(d)研究两种类孤子的传播特性及相互作用。
(5)等谱和一阶非等谱Kaup-Newell(KN)方程族之间的规范变换。(a)简要介绍规范变换的基本概念,引入等谱和一阶非等谱AKNS系统,得到等谱和一阶非等谱KN谱问题之间的规范变换；(b)求得等谱和一阶非等谱KN方程族之间的转化关系,并给出两个方程族的前三个方程；(c)以等谱和一阶非等谱KN系统为例,研究同一系统对应的等谱和非等谱谱问题之间的直接规范变换。
(6)双线性方法及多线性分离变量法在(2+1)维色散长水波方程中的应用。(a)介绍双线性化常用的三种因变量变换及其它形式的因变量变换,以及相应的非线性发展方程的类型；(b)利用多线性分离变量法研究非线性局域激发模式,并列举几种常见局域解的函数形式；(c)对(2+1)维色散长水波方程进行Painleve分析,可得Painleve展开存在单奇异流形和双奇异流形展开两种形式,进而得到两种不同形式的因变量变换；(d)利用两种因变量变换,分别将方程双线性化和线性化,进而求得方程的解析孤子解及非线性局域激发模式,并通过绘图探讨孤子的传播特性和相互作用。
Soliton is an important branch of nonlinear science. It is widely used in mathematics and physics, and is also the theoretical foundation in financial field for studying the characteristics of market evolution. Therefore, the study of soliton theory is of great significance.
This dissertation analytically investigates the integrability and soliton interaction mechanism in nonlinear systems. Through the analysis on some coupled and variable-coefficient nonlinear evolution equations (NLEEs) or nonlinear hierarchies, a series of results are derived, such as the integrable properties, analytical soliton solutions, Hamil-tonian structure and Liouville integrability. This dissertation mainly includes the fol-lowing six parts:
(1) The construction of Darboux transformation (DT) and its applications in the cou-pled NLEEs.(a) The construction of DT for isospectral and nonisospectral integrable systems is investigated, taking the Hirota-Maxwell-Bloch (H-MB) equation, generalized inhomogeneous H-MB equation and inhomogeneous coupled NLS equation for example;(b) Based on the obtained one-and two-soliton solutions of the H-MB equation via the DT, the production mechanism, propagation characteristics and soliton interactions are investigated. The analysis is made to probe the influence of inhomogeneities in the generalized inhomogeneous H-MB equation on the soliton propagation and interac-tions via some figures. With the corresponding parameters of group velocity dispersion, self-phase modulation, cross-phase modulation and gain/loss under control, the soliton interaction mechanism and its potential applications of the inhomogeneous coupled NLS equations are discussed;(c) Via the Painleve analysis, the integrable conditions for the generalized inhomogeneous H-MB equation are obtained. Based on the Ablowitz-Kaup-Newell-Segur (AKNS) system, the Lax pairs associated with the H-MB equations and generalized inhomogeneous H-MB equations are constructed. With the2×2isospectral AKNS spectral problem generalized to the3×3nonisospectral case, the Lax pair for the inhomogeneous coupled NLS equations is derived.
(2) The construction of N-fold DT and asymptotic analysis of the generalized in-homogeneous H-MB equation.(a) The N-fold DT for the generalized inhomogeneous H-MB equation is derived, as well as one-, two-and three-soliton solutions, which can be compiled into the determinant forms;(b) Soliton propagation characteristics and in-teractions with different parameters are discussed, and energy redistribution between the two solitons in their interactions is observed;(c) The asymptotic analysis is used to discuss some physical quantities before and after soliton interactions, such as the energy, amplitude, width, velocity and initial phase;(d) The first three conservation laws of the equation are obtained.
(3) Nonlinear hierarchies and infinite conservation laws.(a) By taking KdV and AKNS hierarchies for example, the processes of constructing the nonlinear hierarchies with the Lax pair in operator or matrix forms are introduced;(b) By taking KdV and AKNS systems for example, the construction processes of the infinite conservation laws with the Lax pair in operator or matrix forms are investigated, and the infinite conservation laws for H-MB equation are also obtained;(c) Based on the generalized discrete spectral problem, the nonlinear hierarchy and infinite conservation laws of the discrete system are derived.
(4) Hamiltonian structure, Darboux transformation and new soliton-like solutions of the Jaulent-Miodek (JM) hierarchy;(a) Based on the JM spectral problem, the integrable JM hierarchy is obtained, and two types of DTs of the hierarchy are constructed;(b) Via the symplectic-cosymplectic factorization, the Hamiltonian structure of JM hierarchy is obtained, and the hierarchy is proved to be completely integrable in the Liouville sense;(c) Through the two type of DTs, new soliton-like solutions are derived, which are all composed of the shock wave and bell-shaped soliton;(d) Propagation characteristics and interactions of the obtained solitons are graphically discussed.
(5) Gauge transformation between the isospectral and first-order nonisospectral Kaup-Newell (KN) hierarchies.(a) The basic concept of gauge transformation is briefly intro-duced. The gauge transformation between the isospectral and first-order nonisospectral KN spectral problems is derived by introducing the isospectral and first-order non-isospectral AKNS spectral problems;(b) The equivalence between the isospectral and first-order nonisospectral KN hierarchies, as well as the first three representative equa-tions, is given;(c) By taking the isospectral and first-order nonisospectral KN systems for example, the direct gauge transformation between the corresponding isospectral and nonisospectral problems of the same system is investigated.
(6) The applications of the bilinear method and multi-linear variable separation ap-proach in (2+1) dimensional dispersive long wave (DLW) equation.(a) Three types of the dependent variable transformations used in the bilinearization, as well as other de-pendent variable transformations, are introduced, and the corresponding types of NLEEs are also presented;(b) The multi-linear variable separation approach is used to investi-gate nonlinear localized excitations, and several common functions for localized solutions are briefly introduced;(c) The Painleve analysis is performed on the (2+1) dimensional DLW equation, and two types of expansion, i.e., one and two singular manifolds, are found. Correspondingly, two different forms of the dependent variable transformation are derived;(d) Via the two types of the dependent variable transformations, the DLW equation is respectively bilinearized and linearized, and further the analytic soliton solu-tions and nonlinear localized excitations are obtained correspondingly. Besides, figures are made to probe the soliton propagation characteristics and interactions.
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