The initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line

TL;DR

This paper investigates the well-posedness of the biharmonic Schrödinger equation on the half-line with boundary conditions, employing the Fokas method, Fourier analysis, and Strichartz estimates to handle linear and nonlinear cases.

Contribution

It introduces a representation formula for the linear problem using the Fokas method and proves well-posedness for nonlinear models with low regularity data.

Findings

Representation formula for linear problem solutions

Local and global well-posedness results

Strichartz estimates for low regularity solutions

Abstract

We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the \emph{unified transform method}). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.