# Emergence of Calogero family of models in external potentials: Duality, Solitons and Hydrodynamics

## TL;DR

This paper introduces a first-order formulation of the Calogero model in external potentials, revealing duality, soliton solutions, and hydrodynamic descriptions, thus advancing understanding of its integrability and collective behaviors.

## Contribution

It presents a novel first-order generating function approach to derive dual forms and soliton solutions for Calogero models in various external potentials, including quartic, trigonometric, and hyperbolic.

## Key findings

- Derived dual forms for Calogero models in multiple external potentials.
- Obtained explicit soliton solutions and demonstrated their integrability.
- Developed a fluid mechanical formulation with stable soliton solutions.

## Abstract

We present a first-order formulation of the Calogero model in external potentials in terms of a generating function, which simplifies the derivation of its dual form. Solitons naturally appear in this formulation as particles of negative mass. Using this method, we obtain the dual form of Calogero particles in external quartic, trigonometric and hyperbolic potentials, which were known to be integrable but had no known dual formulation. We derive the corresponding soliton solutions, generalizing earlier results for the harmonic Calogero system, and present numerical results that demonstrate the integrable nature of the soliton motion. We also give the collective fluid mechanical formulation of these models and derive the corresponding fluid soliton solutions in terms of meromorphic fields, commenting on issues of stability and integrability.