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An explicit formula for the cubic Szegő equation


Authors: Patrick Gérard and Sandrine Grellier
Journal: Trans. Amer. Math. Soc. 367 (2015), 2979-2995
MSC (2010): Primary 37K15; Secondary 47B35
DOI: https://doi.org/10.1090/S0002-9947-2014-06310-1
Published electronically: September 5, 2014
MathSciNet review: 3301889
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Abstract: We derive an explicit formula for the general solution of the cubic Szegő equation and of the evolution equation of the corresponding hierarchy. As an application, we prove that all the solutions corresponding to finite rank Hankel operators are quasiperiodic.


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Patrick Gérard
Affiliation: Université Paris-Sud XI, Laboratoire de Mathématiques d’Orsay, CNRS, UMR 8628, 91405 Orsay Cedex, France and Institut Universitaire de France
Email: Patrick.Gerard@math.u-psud.fr

Sandrine Grellier
Affiliation: Fédération Denis Poisson, MAPMO-UMR 6628, Département de Mathématiques, Université d’Orleans, 45067 Orléans Cedex 2, France
Email: Sandrine.Grellier@univ-orleans.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-06310-1
Keywords: Cubic Szeg\H{o} equation, inverse spectral transform, quasiperiodicity, energy transfer to high frequencies, instability
Received by editor(s): June 19, 2013
Received by editor(s) in revised form: October 6, 2013
Published electronically: September 5, 2014
Additional Notes: Part of this work was completed while the authors were visiting CIRM in Luminy. They are grateful to this institution for its warm hospitality
Article copyright: © Copyright 2014 American Mathematical Society