关于可积系统与超可积系统某些问题的探索
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摘要
本论文主要研究可积系统的可积耦合、超可积系统的自相容源及守恒律、分数阶超可积系统及Bell多项式的应用.
第一章作为绪论,重点介绍可积系统、孤子方程的求解、符号计算的背景与发展现状,阐明本论文的主要工作.
第二章研究可积方程族的两种不同类型的可积耦合.通过构造新的扩展李代数及相应的loop李代数,以其为基元构造新的等谱问题,得到可积方程族,进而求得可积方程族的可积耦合与相应的Hamilton结构；从耦合李代数出发,构造方程族耦合的可积耦合,利用二次型恒等式得到其Hamilton结构.
第三章研究超可积系统的自相容源和守恒律.从loop李超代数得到的超KN可积方程族出发,利用源生成理论导出带自相容源的超KN可积方程族及无穷守恒律,然后用同样的方法导出超JM可积方程族的自相容源和守恒律.
第四章以分数阶导数与微分为基础,应用修正的Riemann-Liouville导数给出分数阶超可积系统的生成理论.由此导出分数阶超AKNS方程族,利用分数阶超迹恒等式得到其相应的分数阶超Hamilton结构.
第五章讨论Bell多项式和相关可积性.利用Bell多项式方法和符号计算,通过参数选择将广义KdV方程转化为两类双线性导数方程,从而推得广义KdV方程的N孤子解、双线性Backlund变换、Lax对、Darboux协变Lax对和无穷守恒律,以揭示其相应的可积性质.同时借助Hirota双线性方法和Riemann theta函数得到该方程的拟周期波解.
In this paper, the integrable coupling of integrable hierarchy, the conservation laws and self-consistent sources of the super integrable systems, fractional super integrable systems and the application of Bell polynomial are mainly discussed.
Chapter1is an introduction to review the theoretical background and development of integrable systems, solution of the soliton equation and symbolic computation. The main works of this dissertation are also illustrated.
Chapter2is devoted to study two different kinds of integrable couplings of inte-grable hierarchies. From the constructed new enlarged Lie algebra and its corresponding loop algebra, new isospectral problems are discussed, which give rise integrable hierar-chies, and then the integrable couplings of the integrable systems and the corresponding Hamiltonian structures are obtained; From coupled Lie algebras, a coupling integrable couplings of an equation hierarchy is constructed and its corresponding Hamiltonian structure is also obtained by quadratic-form identity.
Chapter3concentrates on studying the properties of super integrable hierarchy, such as the self-consistent sources and infinitely conservation laws. From super-KN integrable hierarchies based on loop super Lie algebra, we consider the properties of super-KN integrable hierarchy, such as the self-consistent sources and infinitely con-servation laws using the theory of source. The same method can be used to get the self-consistent sources and the infinitely conservation laws of super-JM integrable hier-archy.
Chapter4discusses the generating of the fractional super hierarchy using the modified Riemann-Liouville derivative, based on the theory of fractional derivatives and integrals. Using the theory of the generating of fractional super systems, we generate the fractional super AKNS hierarchy and obtained its fractional super Hamiltonian structure with the help of fractional supertrace identity.
Chapter5deals with Bell polynomials and related integrabilities. Via Bell poly-nomial approach and symbolic computation, the extended Korteweg-de Vries equation is transformed into two kinds of bilinear equations by choosing different coefficients. N-soliton solutions, bilinear Backlund transformation, Lax pair. Darboux covariant Lax pair and infinite conservation laws are constructed to reveal the integrability of the equation. At the same time, based on Hirota bilinear method and Riemann theta function, its quasi-periodic wave solution is also presented.
第一章作为绪论,重点介绍可积系统、孤子方程的求解、符号计算的背景与发展现状,阐明本论文的主要工作.
第二章研究可积方程族的两种不同类型的可积耦合.通过构造新的扩展李代数及相应的loop李代数,以其为基元构造新的等谱问题,得到可积方程族,进而求得可积方程族的可积耦合与相应的Hamilton结构；从耦合李代数出发,构造方程族耦合的可积耦合,利用二次型恒等式得到其Hamilton结构.
第三章研究超可积系统的自相容源和守恒律.从loop李超代数得到的超KN可积方程族出发,利用源生成理论导出带自相容源的超KN可积方程族及无穷守恒律,然后用同样的方法导出超JM可积方程族的自相容源和守恒律.
第四章以分数阶导数与微分为基础,应用修正的Riemann-Liouville导数给出分数阶超可积系统的生成理论.由此导出分数阶超AKNS方程族,利用分数阶超迹恒等式得到其相应的分数阶超Hamilton结构.
第五章讨论Bell多项式和相关可积性.利用Bell多项式方法和符号计算,通过参数选择将广义KdV方程转化为两类双线性导数方程,从而推得广义KdV方程的N孤子解、双线性Backlund变换、Lax对、Darboux协变Lax对和无穷守恒律,以揭示其相应的可积性质.同时借助Hirota双线性方法和Riemann theta函数得到该方程的拟周期波解.
In this paper, the integrable coupling of integrable hierarchy, the conservation laws and self-consistent sources of the super integrable systems, fractional super integrable systems and the application of Bell polynomial are mainly discussed.
Chapter1is an introduction to review the theoretical background and development of integrable systems, solution of the soliton equation and symbolic computation. The main works of this dissertation are also illustrated.
Chapter2is devoted to study two different kinds of integrable couplings of inte-grable hierarchies. From the constructed new enlarged Lie algebra and its corresponding loop algebra, new isospectral problems are discussed, which give rise integrable hierar-chies, and then the integrable couplings of the integrable systems and the corresponding Hamiltonian structures are obtained; From coupled Lie algebras, a coupling integrable couplings of an equation hierarchy is constructed and its corresponding Hamiltonian structure is also obtained by quadratic-form identity.
Chapter3concentrates on studying the properties of super integrable hierarchy, such as the self-consistent sources and infinitely conservation laws. From super-KN integrable hierarchies based on loop super Lie algebra, we consider the properties of super-KN integrable hierarchy, such as the self-consistent sources and infinitely con-servation laws using the theory of source. The same method can be used to get the self-consistent sources and the infinitely conservation laws of super-JM integrable hier-archy.
Chapter4discusses the generating of the fractional super hierarchy using the modified Riemann-Liouville derivative, based on the theory of fractional derivatives and integrals. Using the theory of the generating of fractional super systems, we generate the fractional super AKNS hierarchy and obtained its fractional super Hamiltonian structure with the help of fractional supertrace identity.
Chapter5deals with Bell polynomials and related integrabilities. Via Bell poly-nomial approach and symbolic computation, the extended Korteweg-de Vries equation is transformed into two kinds of bilinear equations by choosing different coefficients. N-soliton solutions, bilinear Backlund transformation, Lax pair. Darboux covariant Lax pair and infinite conservation laws are constructed to reveal the integrability of the equation. At the same time, based on Hirota bilinear method and Riemann theta function, its quasi-periodic wave solution is also presented.
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