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

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. Our lowest regularity is H1/4×L2, which is somewhat far from the naturally expected endpoint L2×H−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 .Nevertheless, we conclude the global well-posedness of the NLS–KdV system in the energy space H1×H1 using our local well-posedness result and three conservation laws discovered by M. Tsutsumi.