The initial-boundary value problem for the Schrödinger–Korteweg–de Vries system on the half-line

Abstract

We prove local well-posedness for the initial-boundary value problem (IBVP) associated to the Schrödinger–Korteweg–de Vries system on right and left half-lines. The results are obtained in the low regularity setting by using two analytic families of boundary forcing operators, one of these families being developed by Holmer to study the IBVP associated to the Korteweg–de Vries equation [The initial-boundary value problem for the Korteweg–de Vries equation, Comm. Partial Differential Equations 31 (2006) 1151–1190] and the other one was recently introduced by Cavalcante [The initial-boundary value problem for some quadratic nonlinear Schrödinger equations on the half-line, Differential Integral Equations 30(7–8) (2017) 521–554] in the context of nonlinear Schrödinger with quadratic nonlinearities.