Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchy
Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchy
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DOI：
10.1088/1751-8121/acc495
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发表时间：
2022-11
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通讯作者：
K. Takasaki
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文献类型：
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作者：
K. Takasaki
用正整数N表示的中长波(ILW)谱系及其推广可以表示为格子KP谱系的约化。这些约化系统继承了格Kp族的可积性。特别地，所有解都可以通过差分算子的因式分解问题来捕获。其中，CP1的等变Gromov-Witten理论的一个特解是从Okounkov和Pandharipande的Dressing算子得到的。这表明与等变的Toda层次结构之间存在隐藏的联系。推广的ILW族也与格Gelfand-Dickey族及其对数流的扩张有关。当系统的一个参数趋于0时，对数流可以从广义ILW系统按比例极限得到。这解释了对数流的起源。等变Toda层级的类似比例限制产生扩展的1D/双格子Toda层级。
The intermediate long wave (ILW) hierarchy and its generalization, labelled by a positive integer N, can be formulated as reductions of the lattice KP hierarchy. The integrability of the lattice KP hierarchy is inherited by these reduced systems. In particular, all solutions can be captured by a factorization problem of difference operators. A special solution among them is obtained from Okounkov and Pandharipande’s dressing operators for the equivariant Gromov-Witten theory of CP1 . This indicates a hidden link with the equivariant Toda hierarchy. The generalized ILW hierarchy is also related to the lattice Gelfand-Dickey hierarchy and its extension by logarithmic flows. The logarithmic flows can be derived from the generalized ILW hierarchy by a scaling limit as a parameter of the system tends to 0. This explains an origin of the logarithmic flows. A similar scaling limit of the equivariant Toda hierarchy yields the extended 1D/bigraded Toda hierarchy.