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 H^{1 / 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 L² × H^{- frac(3, 4) +}. 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.