Quantum Hydrodynamics, the Quantum Benjamin-Ono Equation, and the Calogero Model

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 analogue of BenjaminOno equation. The latter is known to describe internal waves o...

Integrable Hierarchy of the Quantum Benjamin–Ono Equation

A hierarchy of pairwise commuting Hamiltonians for the quantum periodic Benjamin–Ono equation is constructed by using the Lax matrix. The eigenvectors of these Hamiltonians are Jack symmetric functions of infinitely many variables x1, x2, . . .. This construction provides explicit expressions for the Hamiltonians in terms of the power sum symmetric functions pn = x n 1 + x n 2 + · · · and is ba...

Perturbation theory for the Benjamin–Ono equation

We develop a perturbation theory for the Benjamin–Ono (BO) equation. This perturbation theory is based on the inverse scattering transform for the BO equation, which was originally developed by Fokas and Ablowitz and recently refined by Kaup and Matsuno. We find the expressions for the variations of the scattering data with respect to the potential, as well as the dual expression for the variat...

Benjamin-Ono Equation on a Half-Line

Quantum Solitons in the Calogero-Sutherland Model

We show that the single quasi-particle Schrödinger equation for a certain form of one-body potential yields a stationary one soliton solution. The one-body potential is assumed to arise from the self-interacting charge distribution with the singular kernel of the Calogero-Sutherland model. The quasi-particle has negative or positive charge for negative or positive coupling constant of the inter...