Well-posedness for the Schrödinger-KdV system on the half-line

In this paper we obtain improved local well–posedness results for the Schrödinger-KdV system on the half-line. We employ the Laplace–Fourier method in conjunction with the restricted norm method of Bourgain appropriately modified in order to accommodate the bounded operators of the half–line problem. Our result extends the previous local results in [5], [6] and [18] matching the results that Wu, [25], obtained for the real line system. We also demonstrate the uniqueness for the full range of locally well–posed solutions. In addition we obtain global well–posedness on the half–line for the energy solutions with zero boundary data, along with polynomial–in–time bounds for higher order Sobolev norms for the Schrödinger part.