# Duality in a hyperbolic interaction model integrable even in a strong confinement: Multi-soliton solutions and field theory

## TL;DR

This paper introduces the Hyperbolic Calogero model, a classically integrable one-dimensional system with confining potentials, and explores its multi-soliton solutions, dynamics, and field theory description through analytical and numerical methods.

## Contribution

It presents the first formulation of an integrable hyperbolic interaction model with confinement, along with multi-soliton solutions and a field theory framework.

## Key findings

- Multi-soliton solutions are analytically derived.
- Numerical simulations demonstrate soliton collision dynamics.
- Analytic expressions for the time period of dual variables are obtained.

## Abstract

Models that remain integrable even in confining potentials are extremely rare and almost non-existent. Here, we consider a one-dimensional hyperbolic interaction model, which we call as the Hyperbolic Calogero (HC) model. This is classically integrable even in confining potentials (which have box-like shapes). We present a first-order formulation of the HC model in an external confining potential. Using the rich property of duality, we find multi-soliton solutions of this confined integrable model. Absence of solitons correspond to the equilibrium solution of the model. We demonstrate the dynamics of multi-soliton solutions via brute-force numerical simulations. We studied the physics of soliton collisions and quenches using numerical simulations. We have examined the motion of dual complex variables and found an analytic expression for the time period in a certain limit. We give the field theory description of this model and find the background solution (absence of solitons) analytically in the large-N limit. Analytical expressions of soliton solutions have been obtained in the absence of external confining potential. Our work is of importance to understand the general features of trapped interacting particles that remain classically integrable and can be of relevance to the collective behaviour of trapped cold atomic gases as well.