This paper is dedicated to proving the complete integrability of Benjamin--Ono (BO) equation on line when restricted every $N$-soliton manifold, denoted by $\mathcal{U}_N$. We construct generalized action--angle coordinates which establish a real analytic symplectomorphism from $\mathcal{U}_N$ onto some open convex subset $\mathbb{R}^{2N}$ and allow solve quadrature for any such initial datum. ...

Here we continue the study of the initial value problem for the third order Benjamin-Ono equation in the weighted Sobolev spaces Hs γ = H s⋂L2γ , where s > 3, γ ≥ 0. The result is the proof of well-posedness of the afore mentioned problem in Hs γ , s > 3, γ ∈ [0, 1]. The proof involves the use of parabolic regularization, the Riesz-Thorin interpolation theorem and the construction technique of ...

We define a finite Borel measure of Gibbs type, supported by the Sobolev spaces of negative indexes on the circle. The measure can be seen as a limit of finite dimensional measures. These finite dimensional measures are invariant by the ODE’s which correspond to the projection of the Benjamin-Ono equation, posed on the circle, on the first N , N ≥ 1 modes in the trigonometric bases.

A priori estimates and existence of real-valued periodic solutions to the modified Benjamin-Ono equation with initial data in $H^s$ for $s>1/4$ are proved locally time. The approach relies on frequency dependent time localization, after which dispersive properties from Euclidean space recovered. same regularity results cubic derivative nonlinear Schr\"odinger equation.

We consider the Benjamin–Ono equation on real line for initial data in weighted Sobolev spaces. After application of gauge transform, flow is shown to be Lipschitz continuous and present a nonlinear smoothing effect. As consequence, unconditional uniqueness proved.

We prove that the complex-valued modified Benjamin-Ono (mBO) equation is locally wellposed if the initial data φ belongs to Hs for s ≥ 1/2 with ‖φ‖L2 sufficiently small without performing a gauge transformation. Hence the real-valued mBO equation is globally wellposed for those initial datas, which is contained in the results of C. Kenig and H. Takaoka [25] where the smallness condition is not ...