Soliton solutions of Calogero model in harmonic potential

TL;DR

This paper explores soliton solutions in the classical Calogero model with harmonic potential, providing finite particle and hydrodynamic limit solutions that describe localized density and velocity lumps.

Contribution

It introduces soliton solutions as finite-dimensional reductions of the Calogero model, both for finite particles and in the hydrodynamic limit, expanding understanding of integrable systems.

Findings

Soliton solutions exist for the Calogero model with harmonic potential.

Hydrodynamic limit yields continuous density and velocity field solutions.

Solutions describe localized lumps propagating in a nontrivial background.

Abstract

A classical Calogero model in an external harmonic potential is known to be integrable for any number of particles. We consider here reductions which play a role of "soliton" solutions of the model. We obtain these solutions both for the model with finite number of particles and in a hydrodynamic limit. In the latter limit the model is described by hydrodynamic equations on continuous density and velocity fields. Soliton solutions in this case are finite dimensional reductions of the hydrodynamic model and describe the propagation of lumps of density and velocity in the nontrivial background.