Integrable hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation

TL;DR

This paper develops a hydrodynamic framework for the Calogero-Sutherland model, revealing its connection to the bidirectional Benjamin-Ono equation and deriving related reductions and solutions.

Contribution

It introduces a hydrodynamic description of the Calogero-Sutherland liquid linked to the bidirectional Benjamin-Ono equation and explores its reductions and solutions.

Findings

Hydrodynamic equations form a bidirectional Benjamin-Ono equation.

The bidirectional Benjamin-Ono equation is a real reduction of the modified KP hierarchy.

Constructed multi-phase solutions for the equations.

Abstract

We develop a hydrodynamic description of the classical Calogero-Sutherland liquid: a Calogero-Sutherland model with an infinite number of particles and a non-vanishing density of particles. The hydrodynamic equations, being written for the density and velocity fields of the liquid, are shown to be a bidirectional analogue of Benjamin-Ono equation. The latter is known to describe internal waves of deep stratified fluids. We show that the bidirectional Benjamin-Ono equation appears as a real reduction of the modified KP hierarchy. We derive the Chiral Non-linear Equation which appears as a chiral reduction of the bidirectional equation. The conventional Benjamin-Ono equation is a degeneration of the Chiral Non-Linear Equation at large density. We construct multi-phase solutions of the bidirectional Benjamin-Ono equations and of the Chiral Non-Linear equations.