The profile decomposition for the hyperbolic Schr\"odinger equation

TL;DR

This paper establishes the profile decomposition for hyperbolic Schrödinger equations on  in both mass-supercritical and mass-critical cases, extending previous results and employing advanced harmonic analysis techniques.

Contribution

It provides the first profile decomposition results for hyperbolic Schrödinger equations in these regimes, including a new approach with a double Whitney decomposition.

Findings

Profile decomposition proven for  hyperbolic Schrf6dinger equations

Extension of results to mass-critical and mass-supercritical cases

Introduction of a double Whitney decomposition for handling additional scaling symmetry

Abstract

In this note, we prove the profile decomposition for hyperbolic Schr\"odinger (or mixed signature) equations on $\mathbb{R}^2$ in two cases, one mass-supercritical and one mass-critical. First, as a warm up, we show that the profile decomposition works for the ${\dot H}^{\frac12}$ critical problem, which gives a simple generalization of for instance one of the results in Fanelli-Visciglia (2013). Then, we give the derivation of the profile decomposition in the mass-critical case by proving an improved Strichartz estimate. We will use a very similar approach to that laid out in the notes of Killip-Visan (2008), but we are forced to do a double Whitney decomposition to accommodate an extra scaling symmetry that arises in the problem with mixed signature.