Rough solutions for the periodic Schrödinger-Korteweg-de Vries system

Abstract

We prove two new mixed sharp bilinear estimates of Schrödinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schrödinger-Kortweg-de Vries (NLS–KdV) system in the periodic setting with initial data in sobolev spaces H^sxH^k. Our lowest regularity is (s,k)=(1/4,0), which is somewhat far from the naturally expected endpoint (s,k)=(0,-1/2). This is a novel phenomena related to the periodicity condition. Indeed, in the continuous case, Corcho and Linares proved local well-posedness for the natural endpoint (s,k)=(0,-3/4+). Nevertheless, we conclude the global well-posedness of the NLS-KdV system in the energy space (s,k)=(1,1) using our local well-posedness result and three conservation laws discovered by M. Tsutsumi.