Integrable Hierarchy of the Quantum Benjamin-Ono Equation

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...

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...

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...

An Infinite Branching Hierarchy of Time-Periodic Solutions of the Benjamin-Ono Equation

We present a new representation of solutions of the Benjamin-Ono equation that are periodic in space and time. Up to an additive constant and a Galilean transformation, each of these solutions is a previously known, multi-periodic solution; however, the new representation unifies the subset of such solutions with a fixed spatial period and a continuously varying temporal period into a single ne...

Benjamin-Ono Equation on a Half-Line