The profile decomposition for the hyperbolic Schrödinger equation

Abstract

In this note, we prove the profile decomposition for hyperbolic Schrödinger (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 ${\dot{H}}^{\frac{1}{2}}$ critical problem. Then, we give the derivation of the profile decomposition in the mass-critical case based on an estimate of Rogers-Vargas (J. Functional Anal. 241(2) (2006), 212–231).