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Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy. (arXiv:2211.11353v2 [nlin.SI] UPDATED)

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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 $\mathbb{CP}^1$. This indicates a hidden
link with the equivariant Toda hierarchy. The generalized ILW hierarchy is also
related to the lattice Gelfand-Dickey (GD) 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.

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