Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane

Roberto de A. Capistrano-Filho, Márcio Cavalcante and Fernando A. Gallego

Vol. 309 (2020), No. 1, 35–70

DOI: 10.2140/pjm.2020.309.35

Abstract

We deal with the initial-boundary value problem of the biharmonic cubic nonlinear Schrödinger equation in a quarter plane with inhomogeneous Dirichlet–Neumann boundary data. We prove local well-posedness in the low regularity Sobolev spaces by introducing Duhamel boundary forcing operator associated to the linear equation in order to construct solutions in the whole line. With this in hand, the energy and nonlinear estimates allow us to apply the Fourier restriction method, introduced by J. Bourgain, to obtain our main result. Additionally, we discuss adaptations of this approach for the biharmonic cubic nonlinear Schrödinger equation on star graphs.

Keywords

biharmonic Schrödinger equation, initial-boundary value problem, local well-posedness, quarter plane

Mathematical Subject Classification 2010

Primary: 35A07, 35C15, 35G15, 35G30, 35Q55

Milestones

Received: 23 December 2018

Revised: 29 December 2019

Accepted: 25 July 2020

Published: 26 December 2020