Regularity properties of the cubic nonlinear Schr\"odinger equation on the half line

TL;DR

This paper investigates the regularity and well-posedness of the cubic nonlinear Schrödinger equation on the half line, demonstrating smoothing effects, polynomial and exponential growth of solutions, and providing simplified proofs for existing results.

Contribution

It establishes new regularity results for the cubic NLS on the half line, including smoothing effects and growth bounds, with simplified proofs for local well-posedness.

Findings

Nonlinear part is smoother than initial data

Polynomial growth in defocusing case

Exponential growth in focusing case

Abstract

In this paper we study the local and global regularity properties of the cubic nonlinear Schr\"odinger equation (NLS) on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the behavior of higher order Sobolev norms of the solutions. In particular, we prove that the nonlinear part of the cubic NLS on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic cubic NLS \cite{et2} and the cubic NLS on the line \cite{erin}. We also prove that in the defocusing case the norm of the solution grows at most polynomially-in-time while in the focusing case it grows exponentially-in-time. As a byproduct of our analysis we provide a different proof of an almost sharp local wellposedness in $H^s(\R^+)$. Sharp $L^2$ local wellposedness was obtained in \cite{holmer} and \cite{bonaetal}. Our methods simplify some ideas in the wellposedness theory of initial and boundary value problems that were developed in \cite{collianderkenig, holmer,holmer1,bonaetal}.