arXiv 1004.07512010-04-06nlin.SI

Ablowitz-Ladik等级的对称性：I.四势情形

Symmetries for the Ablowitz-Ladik hierarchy: I. Four-potential case

Da-jun Zhang, Shou-ting Chen

备注：: 20 pages

摘要：

本文首先研究了等谱和非等谱四位势Ablowitz-Ladik族的对称性。我们用$u_{n,t}=L^m H^{(0)}$的形式表示这些层次结构，其中$m$是一个任意整数（而不是自然数），$L$是递归运算符。然后利用等谱流和非等谱流的零曲率表示，分别构造了等谱方程族和非等谱方程族的对称性。这些对称分别构成两个无中心Kac-Moody-维拉索罗代数。证明了该等谱方程族的递归算子$L$是遗传的和强对称的。此外，还明确了四位势与二位势Ablowitz-Ladik族之间的关系。四位势Ablowitz-Ladik族中的偶数阶成员及其对称性和代数结构可归结为二位势情形。该约简保持了代数结构的不变性，两种可能情形的递归算子变为$L^2$。

英文摘要：

In the paper we first investigate symmetries of isospectral and non-isospectral four-potential Ablowitz-Ladik hierarchies. We express these hierarchies in the form of $u_{n,t}=L^m H^{(0)}$, where $m$ is an arbitrary integer (instead of a nature number) and $L$ is the recursion operator. Then by means of the zero-curvature representations of the isospectral and non-isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non-isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac-Moody-Virasoro algebras. The recursion operator $L$ is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four-potential and two-potential Ablowitz-Ladik hierarchies. The even order members in the four-potential Ablowitz-Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two-potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes $L^2$.

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