# KP hierarchy and trigonometric Calogero-Moser hierarchy

## TL;DR

This paper explores the deep connection between the KP hierarchy's trigonometric solutions and the Calogero-Moser model, revealing that pole dynamics correspond to higher Hamiltonians in the hierarchy.

## Contribution

It extends the known pole-particle correspondence from solutions to the entire hierarchy, linking KP evolution to Hamiltonians of the trigonometric Calogero-Moser system.

## Key findings

- Pole dynamics match Calogero-Moser Hamiltonians
- Hierarchy evolution governed by linear combinations of Hamiltonians
- Extension of pole-particle correspondence to hierarchy level

## Abstract

We consider trigonometric solutions of the KP hierarchy. It is known that their poles move as particles of the Calogero-Moser model with trigonometric potential. We show that this correspondence can be extended to the level of hierarchies: the evolution of the poles with respect to the $k$-th hierarchical time of the KP hierarchy is governed by a Hamiltonian which is a linear combination of the first $k$ higher Hamiltonians of the trigonometric Calogero-Moser hierarchy.