— In this paper we construct a Gibbs measure for the derivative Schrödinger equation on the circle. The construction uses some renormalisations of Gaussian series and Wiener chaos estimates, ideas which have already been used by the second author in a work on the Benjamin-Ono equation.

whereH is the Hilbert transform operator defined (on the spaces C(R : H), σ ∈ R) by the Fourier multiplier −i sgn(ξ). The Benjamin–Ono equation is a model for one-dimensional long waves in deep stratified fluids ([1] and [16]) and is completely integrable. The initial-value problem for this equation has been studied extensively for data in the Sobolev spaces H r (R), σ ≥ 0. It is known that the...

We prove the discontinuity for the weak L(T)-topology of the flowmap associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in Hs(T) as soon as s < 0 and thus completes exactly the well-posedness result obtained in [12]. AMS Subject Classification : 35B20, 35Q53.

In this paper, we prove the asymptotic stability of the family of solitons of the Benjamin-Ono equation in the energy space. The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of [16], [18]. As a corollary of the proofs, we obtain the asymptotic stability of exact multi-solitons.

We obtain conservation laws at negative regularity for the Benjamin-Ono equation on line and circle. These conserved quantities control $H^s$ norm of solution $-\frac{1}{2} < s 0$. The are obtained from a study perturbation determinant associated to Lax pair equation.

In this work, the dispersive shock wave (DSW) solution of a Boussinesq Benjamin–Ono (BBO) equation, standard equation with dispersion replaced by nonlocal dispersion, is derived. This DSW derived using two methods, fitting and from simple Whitham modulation equations for BBO equation. The first these yields edges DSW, while second complete solution. As could not be set in Riemann invariant form...

We study the initial value problem associated to the dispersion generalized Benjamin-Ono equation. Our aim is to establish well posedness results in weighted Sobolev spaces and to deduce from them some sharp unique continuation properties of solutions to this equation. In particular, we shall establish optimal decay rate for the solutions of this model. RÉSUMÉ. Nous étudions le problème de Cauc...

We prove that the Benjamin–Ono initial-value problem is locally well-posed for small data in the Banach spaces H̃σ(R), σ ≥ 0, of complex-valued Sobolev functions with special low-frequency structure.