We prove the local well posedness of the Benjamin-Ono equation and the generalized Benjamin-Ono equation in H(T). This leads to a global wellposedness result in H(T) for the 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...

We study the interaction of suitable small and high frequency waves evolving by the flow of the Benjamin-Ono equation. As a consequence, we prove that the flow map of the Benjamin-Ono equation can not be uniformly continuous on bounded sets of H s (R) for s > 0.

and many authors have studied the existence and uniqueness of global solution, the blowup of the solution (see [–] and the references therein). WhenM is not a constant function, equation (.)without the damping and source terms is often called a Kirchhoff-type wave equation; it has first been introduced by Kirchhoff [] in order to describe the nonlinear vibrations of an elastic string. When...

The interacting soliton equation of the Benjamin-Ono equation is found to be st = Hsxx − csx − isxH(s−1sx) − is−1sxH(sx), where H is the Hilbert transform. It is integrable in the sense that it has infinitely many commuting symmetries and conservation laws in involution. Only solitons with speed specified by the spectral parameter c emerge from its time evolution. Since it is connected to the B...

In this work we are interested in the study of controllability and stabilization of the linearized Benjamin-Ono equation with periodic boundary conditions, which is a generic model for the study of weakly nonlinear waves with nonlocal dispersion. It is well known that the Benjamin-Ono equation has infinite number of conserved quantities, thus we consider only controls acting in the equation suc...