We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in H s (R) for any regularity s > − 1 2 . Well-posedness has long been known for s ≥ 0, see [51], but not previously for any s < 0. The scaling-critical value s = − 1 2 is necessarily excluded here, since instantaneous norm inflation is known to occur [11,38,46].We also prove (in a parallel fashion) well-posedness of the real-and complexvalued modified Korteweg-de Vries equations in H s (R) for any s > − 1 2 . The best regularity achieved previously was s ≥ 1 4 ; see [15,24,32,38]. An essential ingredient in our arguments is the demonstration of a local smoothing effect for both equations, with a gain of derivatives matching that of the underlying linear equation. This in turn rests on the discovery of a oneparameter family of microscopic conservation laws that remain meaningful at this low regularity.

Abstract. In this paper we consider the long time behavior of solutions to the modified Korteweg-de Vries equation on R. For sufficiently small, smooth, decaying data we prove global existence and derive modified asymptotics without relying on complete integrability. We also consider the asymptotic completeness problem. Our result uses the method of testing by wave packets, developed in the work of Ifrim and Tataru on the 1d cubic nonlinear Schrödinger and 2d water wave equations.

Abstract. In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match our earlier infinite depth result in [8].

We show that for small, localized initial data there exists a global solution to the KP-I equation in a Galilean-invariant space using the method of testing by wave packets.