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Various aspects of the dynamics of the cubic Szegö solutions
Post-edited
Authors : Grellier, Sandrine (Author of the conference)
CIRM (Publisher )
35B15 - Almost and pseudo-almost periodic solutions of PDE
35B40 - Asymptotic behavior of solutions of PDE
35Q55 - NLS-like equations (nonlinear Schrödinger)
37K15 - Integration of completely integrable systems by inverse spectral and scattering methods
47B35 - Toeplitz operators, Hankel operators, Wiener-Hopf operators
CIRM (Publisher )
Abstract :
The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.
MSC Codes : This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.
35B15 - Almost and pseudo-almost periodic solutions of PDE
35B40 - Asymptotic behavior of solutions of PDE
35Q55 - NLS-like equations (nonlinear Schrödinger)
37K15 - Integration of completely integrable systems by inverse spectral and scattering methods
47B35 - Toeplitz operators, Hankel operators, Wiener-Hopf operators
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 13/07/17 Conference Date : 05/07/2017 Subseries : Research talks arXiv category : Analysis of PDEs ; Classical Analysis and ODEs ; Complex Variables Mathematical Area(s) : PDE ; Dynamical Systems & ODE Format : MP4 (.mp4) - HD Video Time : 00:56:37 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2017-07-04_Grellier.mp4 
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Information on the Event
Event Title : Asymptotic analysis of evolution equations / Analyse asymptotique des équations d'évolutionEvent Organizers : Burq, Nicolas ; Delort, Jean-Marc ; Gérard, Patrick ; Koch, Herbert ; Thomann, Laurent Dates : 03/07/17 - 07/07/17 Event Year : 2017 Event URL : http://conferences.cirm-math.fr/1546.html
Citation Data
DOI : 10.24350/CIRM.V.19192503Cite this video as: Grellier, Sandrine (2017). Various aspects of the dynamics of the cubic Szegö solutions. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19192503 URI : http://dx.doi.org/10.24350/CIRM.V.19192503 |
See Also
- [Multi angle] Geometric heat flows and caloric gauges / Author of the conference Tataru, Daniel.
- [Multi angle] On stability of type II blow up solutions for the critical nonlinear wave equation / Author of the conference Krieger, Joachim.
- [Multi angle] High frequency back reaction for the Einstein equations / Author of the conference Huneau, Cécile.
- [Multi angle] Almost sure scattering for the energy-critical Schrödinger equation in 4D with radial data / Author of the conference Visan, Monica.
Bibliography
- Gérard, P., & Grellier, S. (2017). The cubic Szegö equation and Hankel operators. Astérisque 389. Paris: Société Mathématique de France - http://dx.doi.org/10.24033/ast.1014
- Gérard, P., & Grellier, S. (2016). On the growth of Sobolev norms for the cubic Szegö equation. Séminaire Laurent Schwartz. EDP et Applications 2014-2015, Exp. No. 11 - http://dx.doi.org/10.5802/slsedp.70
