Stability of periodic waves in Hamiltonian PDEs - Institut Camille Jordan Accéder directement au contenu
Communication Dans Un Congrès Année : 2013

Stability of periodic waves in Hamiltonian PDEs

Sylvie Benzoni-Gavage
Pascal Noble
Luis Miguel Miguel Rodrigues
  • Fonction : Auteur
  • PersonId : 1022355

Résumé

Partial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for these waves is still in its infancy though. The issue has been tackled by various means. Of course, it is always possible to address stability from the spectral point of view. However, the link with nonlinear stability ~-~in fact, \emph{orbital} stability, since we are dealing with space-invariant problems~-~, is far from being straightforward when the best spectral stability we can expect is a \emph{neutral} one. Indeed, because of the Hamiltonian structure, the spectrum of the linearized equations cannot be bounded away from the imaginary axis, even if we manage to deal with the point zero, which is always present because of space invariance. Some other means make a crucial use of the underlying structure. This is clearly the case for the variational approach, which basically uses the Hamiltonian -~or more precisely, a constrained functional associated with the Hamiltonian and with other conserved quantities~- as a Lyapunov function. When it works, it is very powerful, since it gives a straight path to orbital stability. An alternative is the modulational approach, following the ideas developed by Whitham almost fifty years ago. The main purpose here is to point out a few results, for KdV-like equations and systems, that make the connection between these three approaches: spectral, variational, and modulational.
Fichier principal
Vignette du fichier
BNR-AEDP-hal.pdf (289.28 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-00915227 , version 1 (06-12-2013)

Licence

Paternité - Pas d'utilisation commerciale - Partage selon les Conditions Initiales

Identifiants

Citer

Sylvie Benzoni-Gavage, Pascal Noble, Luis Miguel Miguel Rodrigues. Stability of periodic waves in Hamiltonian PDEs. Journées Équations aux Dérivées Partielles, David Lannes, Jun 2013, Biarritz, France. pp.22. ⟨hal-00915227⟩
801 Consultations
251 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More