  We consider the cubic Szego equation i u_t=Pi(|u|^2u) on the real line, with
solutions in the Hardy space on the upper half-plane, where Pi is the Szego
projector onto the non-negative frequencies. This equation was recently
introduced by P. Gerard and S. Grellier as a toy model for totally
non-dispersive evolution equations. We show that the only traveling waves are
rational functions with one simple pole. Moreover, they are shown to be
orbitally stable, in contrast to the situation of the circle S^1 studied by the
above authors, where some traveling waves were shown to be unstable.
