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

Abstract

In this paper we obtain improved local well-posedness results for the Schr\"odinger-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 [6], [7] and [21] matching the results that Wu, [28], 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 Schr\"odinger part.