Traveling waves for the cubic Szegő equation on the real line

Abstract

We consider the cubic Szegő equation $i\partial tu=\Pi \left(|u{|}^{2}u\right)$ in the Hardy space $L_{+}^2\left(ℝ\right)$ on the upper half-plane, where $\Pi$ is the Szegő projector. It was first introduced by Gérard and Grellier as a toy model for totally nondispersive evolution equations. We show that the only traveling waves are of the form $C∕\left(x-p\right)$, where $p\in ℂ$ with $Imp<0$. Moreover, they are shown to be orbitally stable, in contrast to the situation on the unit disk where some traveling waves were shown to be unstable.