Explicit formula for the solution of the cubic Szego equation on the real line and its applications

Explicit formula for the solution of the cubic Szego equation on the real line and its applications

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Oana Pocovnicu, Universite Paris-Sud, Orsay
Fine Hall 314

In this talk we consider the cubic Szego equation: $i u_t = Pi (|u|^2u)$ on the real line, where $Pi$ is the Szeg? projector on non-negative frequencies.  This equation was introduced as a model of a non-dispersive Hamiltonian equation.  Like 1-d cubic NLS and KdV, it is known to be completely integrable in the sense that it possesses a Lax pair structure. As a consequence, it turns out that a whole class of finite dimensional manifolds, consisting of rational functions, is invariant under the flow of the Szeg? equation.First, we present an explicit formula for the solutions.  Then, as an application, we prove soliton resolution in $H^s, s>0$, for "generic" data.  Namely, we show that solutions with generic rational functions as initial data can be decomposed in $H^s$, for all $s>0$, into a sum of solitons plus a remainder, as time tends to infinity. As for non-generic data, we construct an example for which soliton resolution holds only in $H^s, 0< s < 1/2$, while the high Sobolev norms grow to infinity over time. As a second application, we construct explicit generalized action-angle coordinates on the manifolds of generic rational functions. In particular, this shows that most trajectories of the Szeg? equation spiral around Lagrangian toroidal cylinders (the non-compact generalizations of the Lagrangian tori in the Liouville-Arnold theorem).