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Journées équations aux dérivées partielles

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Zaher Hani
Out-of-equilibrium dynamics and statistics of dispersive PDE
Journées équations aux dérivées partielles (2016), Exp. No. 5, 12 p., doi: 10.5802/jedp.646
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Mots clés: Modified scattering, nonlinear Schrödinger equation, wave guide manifolds, energy cascade, weak turbulence

Résumé - Abstract

The purpose of this note is to report on some recent advances in the study of out-of-equilibrium behavior of dispersive PDE. One can address this problematic from two different perspectives: a dynamical systems one, and a statistical physics one. The dynamical systems perspective corresponds to constructing solutions exhibiting “energy cascade” between scales, whereas the statistical physics perspective corresponds to deriving effective equations for the dynamics under some “macroscopic limits” in what is often called wave turbulence theory. The rigorous justification of this theory is an outstanding open problem from a rigorous mathematical point of view, and we will touch on it here. We shall discuss some recent attempts to better understand both of the above perspectives.

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