Wave Turbulence and Complete Integrability

  • Patrick GérardEmail author
Part of the Fields Institute Communications book series (FIC, volume 83)


The most famous completely integrable PDEs—KdV and 1D cubic NLS—do not display any kind of wave turbulence, since the conservation laws control high regularity, making impossible the long time appearance of small scales. In this course, I will discuss a new type of integrable infinite-dimensional system, posed on functions of one variable, allowing dramatic growth of high Sobolev norms. The analysis is connected to the solution of an inverse spectral problem for Hankel operators from classical analysis. I will also discuss how this phenomenon can be exported to some Hamiltonian PDEs which can be seen as perturbations of this new type of integrable system.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRSUniversité Paris-SaclayOrsayFrance

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