Symmetries for the Ablowitz-Ladik hierarchy: I. Four-potential case
Symmetries for the Ablowitz-Ladik hierarchy: I. Four-potential case
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发表时间：
2010-04
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通讯作者：
Da‐jun Zhang;Shou-Ting Chen
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作者：
Da‐jun Zhang;Shou-Ting Chen
本文首先研究了等谱和非等谱四势Ablowitz-Ladik层次的对称性。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，分别构造了等谱流和非等谱流的对称性。这些对称分别形成两个无中心的Kac-Moody-Virasoro代数。证明了该等谱方程递归算子$L$具有遗传性和强对称性。此外，我们还明确了四势和二势Ablowitz-Ladik层次之间的关系。四势Ablowitz-Ladik阶中的偶阶元及其对称性和代数结构可以简化为二势情形。代数结构的约简保持不变，两种可能情况的递归算子变为$L^2$。
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 $L^2$.