Harmonic analysis and partial differential equations / Analyse harmonique et équations aux dérivées partielles

Collection Harmonic analysis and partial differential equations / Analyse harmonique et équations aux dérivées partielles

Organizer(s) Bernicot, Frédéric ; Martell, José Maria ; Monniaux, Sylvie ; Portal, Pierre

Date(s) 10/06/2024 - 14/06/2024

linked URL https://conferences.cirm-math.fr/2979.html

00:00:00 / 00:00:00

3 5

Traveling waves for a family of Szegö equations

By Sandrine Grellier

About fifteen years ago, Patrick Gérard and I introduced the cubic Szegö equation$$\begin{aligned}i \partial_{t} u & =\Pi\left(|u|^{2} u\right), \quad u=u(x, t), \quad x \in \mathbb{T}, t \in \mathbb{R} \u(x, 0) & =u_{0}(x) .\end{aligned}$$Here $\Pi$ denotes the Szegö projector which maps $L^{2}(\mathbb{T})$-functions into the Hardy space of $L^{2}(\mathbb{T})$-traces of holomorphic functions in the unit disc. It turned out that the dynamics of this equation were unexpected. This motivated us to try to understand whether the cubic Szegö equation is an isolated phenomenon or not. This talk is part of this project. We consider a family of perturbations of the cubic Szegö equation and look for their traveling waves. Let us recall that traveling waves are particular solutions of the form$$u(x, t)=\mathrm{e}^{-i \omega t} u_{0}\left(\mathrm{e}^{-i c t} x\right), \quad \omega, c \in \mathbb{R}$$We will explain how the spectral analysis of some operators allows to characterize them. From joint works with Patrick Gérard.

Information about the video

Date of recording 11/06/2024
Date of publication 10/07/2024
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Luca Recanzone
Format MP4

Citation data

DOI 10.24350/CIRM.V.20189203
Cite this video Grellier, Sandrine (11/06/2024). Traveling waves for a family of Szegö equations. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20189203
URL https://dx.doi.org/10.24350/CIRM.V.20189203

Domain(s)

Bibliography

GÉRARD, Patrick et GRELLIER, Sandrine. A survey of the Szegő equation. Science China Mathematics, 2019, vol. 62, p. 1087-1100. - http://dx.doi.org/10.1007/s11425-018-9497-0
GÉRARD, Patrick et GRELLIER, Sandrine. The cubic Szegő equation. In : Annales scientifiques de l'école Normale Supérieure. 2010. p. 761-810. - https://doi.org/10.24033/asens.2133

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