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, being one of these family developed by Holmer to study the IBVP associated to the Korteweg-de Vries equation (Communications in Partial Differential Equations, 31 (2006)) and the other family one was recently introduced by Cavalcante (Differential and Integral Equations, 30 (2017)) in the context of nonlinear Schrödinger with quadratic nonlinearities.

Well-posedness and lower bounds of the growth of weighted norms for the Schrödinger–Korteweg–de Vries interactions on the half-line

“…So, in this paper we are interested in to establish a local theory for high regularity initial data, including the energy space, and consequently the possibility to show the existence of global solutions in this space under suitable assumptions for the boundary conditions. In this address the same approach used in [10] can not be applied due the difficulty to estimate the boundary forcing operators in high regularity in the context of Bourgain spaces. So, to establish well-posedness we will follow closely the ideas developed in the works [5] and [14].…”

“…The UTM method provides a generalization of the Inverse Scattering Transform (IST) method from initial value problems to IBVPs. The classical method based on the Laplace transform was used successfully in the works [5,6,14,15] [8,10,24,25]. On the other hand, in [16], Faminskii used an approach based on the investigation of special solutions of a "boundary potential" type for solution of linearized KdV equation in order to obtain global results for the IBVP associated to the KdV equation on the half-line with more general boundary conditions.…”

“…Here we are interesting in the dynamics of the solutions in the energy space H 1 (R ± ) × H 1 (R ± ), so we need to justify a local theory for the IBVP (1.1) in high regularity assumptions. The estimates on Bourgain spaces of the boundary operator given in [10] do not reach high regularity, so we need to use another method. In this direction, the crucial first step is to obtain an appropriate operators that solve the following linear versions of the IBVPs:…”

“…The following mixed bilinear estimates can be obtained without major difficulties following the same ideas in [10]. Lemma 3.7.…”

“…More recently, in [10], the IBVP (1.1) and (1.2) was studied for initial-boundary data with the following configuration: . This setting was motivated by the following smoothing effects given in [28], namely…”