Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions

Abstract

An interaction equation of the capillary-gravity wave is considered. We show that the Cauchy problem of the coupled Schrödinger-KdV equation, (itu + x2u -αvu + γ|u|2u, x ∈ R, tv + x3 + xv2 =βx(|u|2), u(x, 0) = u0(x), v(x, 0) = v0(x), is locally well-posed for weak initial data U0 × v0 ε L2(R) × L21/2(R) × H-1/2 (R). We apply the analogous method for estimating the nonlinear coupling terms developed by Bourgain and refined by Kenig, Ponce, and Vega. © 1997 American Mathematical Society.