We consider an initial value problem for a quadratically nonlinear inviscid BurgersHilbert equation that models the motion of vorticity discontinuities. We use a normal form transformation, which isâ€¦ (More)

Abstract. This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates.â€¦ (More)

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates, and proveâ€¦ (More)

This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problemâ€¦ (More)

This article is concerned with the small data problem for the cubic nonlinear SchrÃ¶dinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simplerâ€¦ (More)

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 localâ€¦ (More)

We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a modified energy method to prove theâ€¦ (More)

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.

This paper derives a 2D, quasi-linear SchrÃ¶dinger equation in Lagrangian coordinates that describes the effects of weak pressure gradients on large amplitude inertial oscillations in a rotatingâ€¦ (More)