Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy

TL;DR

This paper explores the generalized ILW hierarchy, its solutions, and its connection to the lattice Gelfand-Dickey hierarchy, revealing how logarithmic flows emerge from scaling limits and linking to equivariant Gromov-Witten theory.

Contribution

It introduces a generalized ILW hierarchy as a reduction of lattice KP, connects solutions to Gromov-Witten theory, and explains the origin of logarithmic flows via scaling limits.

Findings

Solutions captured by difference operator factorization

Link between ILW hierarchy and equivariant Gromov-Witten theory

Logarithmic flows derived from scaling limits

Abstract

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.