In this article, we establish new results about the ill-posedness of the Cauchy problem for the modified Korteweg-de Vries and the defocusing modified Korteweg-de Vries equations, in the periodic case. The lack of local well-posedness is in the sense that the dependence of solutions upon initial data fails to be continuous. We also develop a method for obtaining ill-posedness results in the per...

In this paper we propose a new approach to prove the local well-posedness of the Cauchy problem associated with strongly non resonant dispersive equations. As an example we obtain unconditional well-posedness of the Cauchy problem below H for a large class of one-dimensional dispersive equations with a dispersion that is greater or equal to the one of the Benjamin-Ono equation. Since this is do...

We study the initial value problem associated to the BenjaminOno equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces Zs,r = Hs(R) ∩L2(|x|2rdx), s ∈ R, s ≥ 1 and s ≥ r. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demonstrate that our persistence proper...

We study the higher-order nonlinear dispersive equation ∂tu+ ∂ 2j+1 x u = ∑ 0≤j1+j2≤2j aj1,j2∂ j1 x u∂ j2 x u, x, t ∈ R. where u is a real(or complex-) valued function. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when a0,k 6= 0 for some k > j, in the sense that this equation cann...

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...

and Applied Analysis 3 The following type of Strichartz estimate was used in [19, 20] for the study of the Benjamin-Ono equation. We refer to [12] for the counterpart to the Schrödinger equation. Lemma 5. Let T ≤ 1 and V be a solution to the equation i∂ t V + ΔV = F 1 + F 2 , (t, x) ∈ (0, T) ×R 2 . (15) Then, for δ ∈ R and ε > 0, one has 󵄩󵄩󵄩󵄩 J δ V 󵄩󵄩󵄩󵄩L p T L q ≲ ‖V‖ L ∞ T H δ+1/2+ε + 󵄩󵄩󵄩󵄩F1 󵄩...