CIRM - Videos & books Library - Unresolved problems in the theory of integrable systems

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Auteurs : Zakharov, Vladimir (Auteur de la Conférence)
CIRM (Editeur )

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bounded non-vanishing solutions KdV equation dressing method new bounded solutions NLS equation with random initial data numerics with NLS equation - condensate IC numerics with NLS equation - cnoidal wave IC numerics with NLS equation - soliton gas IC conjecture of integrability of the deep water equation super compact equation compressed fluid solutions additional motion constants rogue wave in simplified framework rogue wave in exact framework formation of drop questions of the audience

Résumé : In spite of enormous success of the theory of integrable systems, at least three important problems are not resolved yet or are resolved only partly. They are the following:
1. The IST in the case of arbitrary bounded initial data.
2. The statistical description of the systems integrable by the IST. Albeit, the development of the theory of integrable turbulence.
3. Integrability of the deep water equations.
These three problems will be discussed in the talk.

Codes MSC :
35Q53 - KdV-like (Korteweg-de Vries) equations
35Q55 - NLS-like equations (nonlinear Schrödinger)
37K10 - Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)
76B15 - Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76Fxx - Turbulence
35C08 - Soliton solutions of PDE
35C07 - Traveling wave solutions of PDE

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https://videos.cirm-math.fr/2017-06-14_Zakharov.mp4

Informations sur la Rencontre

Nom de la rencontre : French-American conference on nonlinear dispersive PDEs / Conférence franco-américaine sur les EDP dispersives non linéaires
Organisateurs de la rencontre : Carles, Rémi ; Holmer, Justin ; Roudenko, Svetlana
Dates : 12/06/17 - 16/06/17
Année de la rencontre : 2017
URL Congrès : http://conferences.cirm-math.fr/1510.html

Données de citation

DOI : 10.24350/CIRM.V.19182503
Citer cette vidéo: Zakharov, Vladimir (2017). Unresolved problems in the theory of integrable systems. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19182503
URI : http://dx.doi.org/10.24350/CIRM.V.19182503

Bibliographie

Dyachenko, S.A., Zakharov, D.V., & Zakharov, V.E. (2016). Primitive potentials and bounded solutions of the KdV equation. Physica D, 333, 148-156 - https://doi.org/10.1016/j.physd.2016.04.002

Zakharov, D.V., & Zakharov, V.E., & Dyachenko, S.A. (2016). Non-periodic one-dimensional ideal conductors and integrable turbulence. Physical Letters A, 380(46), 3881-3885 - https://doi.org/10.1016/j.physleta.2016.09.040

Zakharov, D.V., Dyachenko, S.A., & Zakharov, V.E. (2016). Bounded solutions of KdV and non periodic one-gap potentials in quantum mechanics. Letters in Mathematical Physics, 106(6). 731-740 - http://dx.doi.org/10.1007/s11005-016-0838-6

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