The Schrödinger-Korteweg-de Vries system on the half-line

Abstract

Using the Fokas unified transform method (UTM), the well-posedness of the initial-boundary value problem (ibvp) for the Schrödinger-Korteweg-de Vries system on the half-line is studied for initial data (u0,v0)(x) in spatial Sobolev spaces Hs1(0,∞)×Hs2(0,∞), s1>0, s2>−3/4, and boundary data (g0,h0)(t) in the temporal Sobolev spaces suggested by the time regularity of the Cauchy problem for the corresponding linear system. First, linear estimates in Bourgain spaces are derived by utilizing the Fokas solution formula of the ibvp for the forced linear system. Then, using these and the needed bilinear/trilinear estimates, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space.