半离散AKNS系统的对称与代数结构
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摘要
本文主要研究若干经典的半离散可积模型,系统地建立这些模型的无穷多对称及Lie代数结构.这些模型包括:
·4-位势Ablowitz-Ladik方程族.
·半离散AKNS方程族.
·可积半离散非线性Schr(o|¨)dinger方程.
·半离散mKdV方程.
论文首先研究4-位势Ablowitz-Ladik等谱与非等谱方程族的对称.从4-位势Ablowitz-Ladik谱问题出发得到其方程族,并将方程族表示成un,t = LmH(0),其中L是递推算子, m不再只是自然数,而是定义在整个整数范围上.利用等谱流与非等谱流的零曲率表示构造出等谱与非等谱方程族的两组无限多的对称.这些对称生成的两个无穷维Lie代数是无中心的Kac-Moody-Virasoro代数.证明递推算子L是等谱方程族的遗传强对称,并给出非等谱方程族的一个强对称.此外,我们找到了4-位势Ablowitz-Ladik方程族、对称和Lie代数与2-位势的相关结果之间的关系: 4-位势第偶数个方程及其对称和代数结构对应着2-位势的情况,递推算子的平方L2对应着2-位势的递推算子.
其次,论文考虑与Ablowitz-Ladik方程族有关的系统的对称,即构造半离散AKNS方程族、可积半离散非线性Schr(o|¨)dinger方程和半离散mKdV方程的对称.借助2-位势Ablowitz-Ladik系统,利用对二阶导数的中心差分离散的思想,通过拼接得到半离散AKNS等谱流、非等谱流和递推算子L.在连续极限下半离散AKNS等谱流和非等谱流成为AKNS系统的流,递推算子成为AKNS递推算子的平方.这些等谱流和非等谱流生成一个Lie代数,其代数结构对构造半离散AKNS系统及其约化系统的对称和Lie代数都起着关键的作用.而这个Lie代数与连续系统里的代数结构不同,不再是无中心的Kac-Moody-Virasoro代数.我们根据连续极限理论和引入“阶”的定义来解释这两者之间在连续极限过程中产生的代数变形.向量形式的可积半离散非线性Schr(o|¨)dinger方程族和半离散mKdV方程族可以分别通过对半离散AKNS系统强加约化Rn = -εQn·和Rn =-εQn得到.而(标量形式的)半离散非线性Schr(o|¨)dinger方程族的对称及Lie代数的生成必须满足Lie括号约化的封闭性.此外,通过选取新的等谱流和非等谱流以及递推算子L_0,可以得到另一类半离散AKNS方程族,进而通过约化得到新的半离散非线性Schr(o|¨)dinger和半离散mKdV方程族,作用连续极限后它们也可以分别连续到连续NLS和mKdV系统里的结果.
最后讨论半离散AKNS方程族的双Hamilton结构及其连续极限.通过引入2-位势Ablowitz-Ladik方程族的多Hamilton结构以及递推算子分解中的逆辛算子θ和辛算子J,将θ和J进行不同的线性组合,我们得到半离散AKNS方程族的递推算子的多种逆辛-辛分解,进而给出其双Hamilton结构.而在连续极限的作用下,半离散AKNS方程族的双Hamilton结构可以连续到AKNS方程族的双Hamilton结构.
In this dissertation we mainly investigate symmetries and their Lie algebraic struc-tures for some classical integrable semi-discrete systems, which are
·four-potential Ablowitz-Ladik hierarchies.
·semi-discrete AKNS hierarchies.
·integrable semi-discrete nonlinear Schr¨odinger equation.
·semi-discrete mKdV equation.
We first investigate symmetries of the isospectral and non-isospectral four-potentialAblowitz-Ladik hierarchies. We derive isospectral and non-isospectral equation hierarchiesfrom the four-potential Ablowitz-Ladik spectral problem, and we express them in theform of un,t = LmH(0), where m is an arbitrary integer (instead of a nature number)and L is the recursion operator. Then by means of the zero-curvature representationsof the isospectral and non-isospectral ?ows, we construct two sets of symmetries for theisospectral equation hierarchy as well as non-isospectral equation hierarchy, respectively.The symmetries, respectively, form two centerless Kac-Moody-Virasoro algebras. Therecursion operator L is proved to be hereditary and a strong symmetry for the isospectralequation hierarchy. We also find a strong symmetry operator for each non-isospectral four-potential Ablowitz-Ladik equation. Besides, we make clear for the relation between four-potential and two-potential Ablowitz-Ladik hierarchies together with their symmetriesand algebraic structures. The even order members in the four-potential Ablowitz-Ladikhierarchies together with their symmetries and algebraic structures can be reduced totwo-potential case. The reduction keeps invariant for the algebraic structures and therecursion operator for two potential case becomes L2.
Then we consider symmetries for some systems which are related to the Ablowitz-Ladik hierarchy. We derive symmetries for the semi-discrete AKNS hierarchy, integrablesemi-discrete nonlinear Schro¨dinger hierarchy and semi-discrete mKdV hierarchy. Byvirtue of a central-di?erence discretization for the second order derivative in the con-tinuous NLS equation, we present semi-discrete AKNS isospectral ?ows which consist of positive as well as negative order two-potential AL isospectral flows, non-isospectralflows and their recursion operator, respectively. In continuous limit these flows go to thecontinuous AKNS ?ows and the recursion operator goes to the square of the AKNS re-cursion operartor. These semi-discrete AKNS flows form a Lie algebra which plays a keyrole in constructing symmetries and their algebraic structures for both the semi-discreteAKNS hierarchy and its reduction cases. Structures of the obtained algebras are dif-ferent from those in continuous cases which usually are centerless Kac-Moody-Virasorotype. These algebra deformations are explained through continuous limit and“degree”interms of lattice spacing parameter h. As reduced cases, the vector form of the integrablesemi-discrete nonlinear Schro¨dinger hierarchy and semi-discrete mKdV hierarchy can beobtained under the reduction of R_n = -εQ*n and R_n = -εQ_n, respectively. However,for the scalar form of the integrable semi-discrete nonlinear Schro¨dinger hierarchy, theirsymmetries and Lie algebra must guarantee the closeness of algebraic structure. Besides,we construct another semi-discrete AKNS hierarchy and discuss reduction cases, whichalso go to the continuous AKNS system and reduction system under continuous limit.
Finally, we discuss the bi-Hamiltonian structures and its continuous limit of the semi-discrete AKNS hierarchy. Using the multi-Hamiltonian structures and implectic operatorθand symplectic operator J of the two-potential Ablowitz-Ladik hierarchy, we presentseveral implectic-symplectic factorization of the recursion operator for the semi-discreteAKNS hierarchy with di?erent linear combinations ofθand J. This enables us to discussbi-Hamiltonian structures of the semi-discrete AKNS hierarchy. In continuous limit thesebi-Hamiltonian structures go to the bi-Hamiltonian structure of the continuous AKNSsystem.
·4-位势Ablowitz-Ladik方程族.
·半离散AKNS方程族.
·可积半离散非线性Schr(o|¨)dinger方程.
·半离散mKdV方程.
论文首先研究4-位势Ablowitz-Ladik等谱与非等谱方程族的对称.从4-位势Ablowitz-Ladik谱问题出发得到其方程族,并将方程族表示成un,t = LmH(0),其中L是递推算子, m不再只是自然数,而是定义在整个整数范围上.利用等谱流与非等谱流的零曲率表示构造出等谱与非等谱方程族的两组无限多的对称.这些对称生成的两个无穷维Lie代数是无中心的Kac-Moody-Virasoro代数.证明递推算子L是等谱方程族的遗传强对称,并给出非等谱方程族的一个强对称.此外,我们找到了4-位势Ablowitz-Ladik方程族、对称和Lie代数与2-位势的相关结果之间的关系: 4-位势第偶数个方程及其对称和代数结构对应着2-位势的情况,递推算子的平方L2对应着2-位势的递推算子.
其次,论文考虑与Ablowitz-Ladik方程族有关的系统的对称,即构造半离散AKNS方程族、可积半离散非线性Schr(o|¨)dinger方程和半离散mKdV方程的对称.借助2-位势Ablowitz-Ladik系统,利用对二阶导数的中心差分离散的思想,通过拼接得到半离散AKNS等谱流、非等谱流和递推算子L.在连续极限下半离散AKNS等谱流和非等谱流成为AKNS系统的流,递推算子成为AKNS递推算子的平方.这些等谱流和非等谱流生成一个Lie代数,其代数结构对构造半离散AKNS系统及其约化系统的对称和Lie代数都起着关键的作用.而这个Lie代数与连续系统里的代数结构不同,不再是无中心的Kac-Moody-Virasoro代数.我们根据连续极限理论和引入“阶”的定义来解释这两者之间在连续极限过程中产生的代数变形.向量形式的可积半离散非线性Schr(o|¨)dinger方程族和半离散mKdV方程族可以分别通过对半离散AKNS系统强加约化Rn = -εQn·和Rn =-εQn得到.而(标量形式的)半离散非线性Schr(o|¨)dinger方程族的对称及Lie代数的生成必须满足Lie括号约化的封闭性.此外,通过选取新的等谱流和非等谱流以及递推算子L_0,可以得到另一类半离散AKNS方程族,进而通过约化得到新的半离散非线性Schr(o|¨)dinger和半离散mKdV方程族,作用连续极限后它们也可以分别连续到连续NLS和mKdV系统里的结果.
最后讨论半离散AKNS方程族的双Hamilton结构及其连续极限.通过引入2-位势Ablowitz-Ladik方程族的多Hamilton结构以及递推算子分解中的逆辛算子θ和辛算子J,将θ和J进行不同的线性组合,我们得到半离散AKNS方程族的递推算子的多种逆辛-辛分解,进而给出其双Hamilton结构.而在连续极限的作用下,半离散AKNS方程族的双Hamilton结构可以连续到AKNS方程族的双Hamilton结构.
In this dissertation we mainly investigate symmetries and their Lie algebraic struc-tures for some classical integrable semi-discrete systems, which are
·four-potential Ablowitz-Ladik hierarchies.
·semi-discrete AKNS hierarchies.
·integrable semi-discrete nonlinear Schr¨odinger equation.
·semi-discrete mKdV equation.
We first investigate symmetries of the isospectral and non-isospectral four-potentialAblowitz-Ladik hierarchies. We derive isospectral and non-isospectral equation hierarchiesfrom the four-potential Ablowitz-Ladik spectral problem, and we express them in theform of un,t = LmH(0), where m is an arbitrary integer (instead of a nature number)and L is the recursion operator. Then by means of the zero-curvature representationsof the isospectral and non-isospectral ?ows, we construct two sets of symmetries for theisospectral equation hierarchy as well as non-isospectral equation hierarchy, respectively.The symmetries, respectively, form two centerless Kac-Moody-Virasoro algebras. Therecursion operator L is proved to be hereditary and a strong symmetry for the isospectralequation hierarchy. We also find a strong symmetry operator for each non-isospectral four-potential Ablowitz-Ladik equation. Besides, we make clear for the relation between four-potential and two-potential Ablowitz-Ladik hierarchies together with their symmetriesand algebraic structures. The even order members in the four-potential Ablowitz-Ladikhierarchies together with their symmetries and algebraic structures can be reduced totwo-potential case. The reduction keeps invariant for the algebraic structures and therecursion operator for two potential case becomes L2.
Then we consider symmetries for some systems which are related to the Ablowitz-Ladik hierarchy. We derive symmetries for the semi-discrete AKNS hierarchy, integrablesemi-discrete nonlinear Schro¨dinger hierarchy and semi-discrete mKdV hierarchy. Byvirtue of a central-di?erence discretization for the second order derivative in the con-tinuous NLS equation, we present semi-discrete AKNS isospectral ?ows which consist of positive as well as negative order two-potential AL isospectral flows, non-isospectralflows and their recursion operator, respectively. In continuous limit these flows go to thecontinuous AKNS ?ows and the recursion operator goes to the square of the AKNS re-cursion operartor. These semi-discrete AKNS flows form a Lie algebra which plays a keyrole in constructing symmetries and their algebraic structures for both the semi-discreteAKNS hierarchy and its reduction cases. Structures of the obtained algebras are dif-ferent from those in continuous cases which usually are centerless Kac-Moody-Virasorotype. These algebra deformations are explained through continuous limit and“degree”interms of lattice spacing parameter h. As reduced cases, the vector form of the integrablesemi-discrete nonlinear Schro¨dinger hierarchy and semi-discrete mKdV hierarchy can beobtained under the reduction of R_n = -εQ*n and R_n = -εQ_n, respectively. However,for the scalar form of the integrable semi-discrete nonlinear Schro¨dinger hierarchy, theirsymmetries and Lie algebra must guarantee the closeness of algebraic structure. Besides,we construct another semi-discrete AKNS hierarchy and discuss reduction cases, whichalso go to the continuous AKNS system and reduction system under continuous limit.
Finally, we discuss the bi-Hamiltonian structures and its continuous limit of the semi-discrete AKNS hierarchy. Using the multi-Hamiltonian structures and implectic operatorθand symplectic operator J of the two-potential Ablowitz-Ladik hierarchy, we presentseveral implectic-symplectic factorization of the recursion operator for the semi-discreteAKNS hierarchy with di?erent linear combinations ofθand J. This enables us to discussbi-Hamiltonian structures of the semi-discrete AKNS hierarchy. In continuous limit thesebi-Hamiltonian structures go to the bi-Hamiltonian structure of the continuous AKNSsystem.
引文
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