Integrable hydrodynamics of Calogero–Sutherland model: bidirectional Benjamin–Ono equation

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 analog of the 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 nonlinear equation which appears as a chiral reduction of the bidirectional equation. The conventional Benjamin–Ono equation is a degeneration of the chiral nonlinear equation at large density. We construct multi-phase solutions of the bidirectional Benjamin–Ono equations and of the chiral nonlinear equations.