Sistemi integrabili infinito dimensionali e loro perturbazioni

Bambusi, Dario ; Maspero, Alberto

Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana, Tome 2 (2017), p. 309-326 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

Durante gli ultimi 50 anni, sono stati fatti enormi progressi nella comprensione del comportamento qualitativo di equazioni a derivate parziali non lineari. In modo specifico, l'estensione a questo ambito dei metodi della meccanica Hamiltoniana ha permesso dapprima di capire che esiste un'intera classe di equazioni, chiamate ``integrabili'', le cui soluzioni hanno sempre carattere ricorrente, e successivamente di cominciare a comprendere ciò che avviene quando queste equazioni sono perturbate e danno luogo a sistemi in cui possono coesistere comportamenti regolari e comportamenti turbolenti. Nel nostro articolo, presenteremo alcuni dei risultati di questa teoria, a partire dalle sue origini fino a oggi, e discuteremo alcuni dei più importanti problemi aperti.

Abstract

The last 50 years have seen enourmous advances in the comprehension of the qualitative behaviour of solutions of nonlinear partial differential equations. In particular the extension to this field of the methods of Hamiltonian mechanichs has been the key for the discovery of a full class of equations called ``integrable'', whose solutions always have a recurrent behaviour and has also allowed to shed some light on the solutions of perturbations of integrable equations, which can display both a recurrent and a turbulent behaviour. In this paper we will present some of the results of the theory from its beginning to our days and we will discuss some of the most important open problems.

Détail

Publié le : 2017-12-01

MR 3753847

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