Traveling waves for the cubic Szegő equation on the real line

We consider the cubic Szego equation i partial derivative tu = Pi(vertical bar u vertical bar(2) u) in the Hardy space L-+(2) (R) on the upper half-plane, where Pi is the Szego projector. It was first introduced by Gerard and Grellier as a toy model for totally nondispersive evolution equations. We show that the only traveling waves are of the form C/(x - p), where p epsilon C with Im p <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.