Traveling waves for the cubic Szego equation on the real line

Details Traveling waves for the cubic Szego equation on the real line Traveling waves for the cubic Szego equation on the real line

2010-01-22

arxiv.org/abs/1001.4037v2

Mathematics

Analysis of PDEs

24 pages, added references, small revision of the second part of the proof of Theorem 2.1 (p.8-9)

Scientific paper

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.

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