Well-Posedness of the Nonlinear Schrödinger Equation on the Half-Plane

2018/10/04 by Himonas, A. Alexandrou, Mantzavinos, Dionyssios
#35G16 #35G31 #35Q55 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.02395

Abstract

The initial-boundary value problem (IBVP) for the nonlinear Schrödinger (NLS) equation on the half-plane with nonzero boundary data is studied by advancing a novel approach recently developed for the well-posedness of the cubic NLS on the half-line, which takes advantage of the solution formula produced by the unified transform of Fokas for the associated linear IBVP. For initial data in Sobolev spaces on the half-plane and boundary data in Bourgain spaces arising naturally when the linear IBVP is solved with zero initial data, the present work provides a local well-posedness result for NLS initial-boundary value problems in higher dimensions.

Related

• The nonlinear Schrödinger equation on the half-space
• The Robin and Neumann problems for the nonlinear Schrödinger equation on the half-plane
• The initial-boundary value problem for some quadratic nonlinear Schrödinger equations on the half-line
• Low-Regularity Solutions of the Nonlinear Schrödinger Equation on the Spatial Quarter-Plane
• Local well-posedness of the higher order nonlinear Schrödinger equation on the half-line: single boundary condition case