Les équations aux dérivées partielles munies d’une structure hamiltonnienne sont connues pour admettre des familles entières d’ondes progressives périodiques. C’est le cas pour l’équation de Korteweg–de Vries et de nombreux autres modèles plus ou moins classiques. L’étude de la stabilité de ces ondes en est cependant encore à ses balbutiements. Plusieurs approches sont possibles. L’une d’elles est bien sûr l’analyse spectrale des équations linéarisées. Toutefois, le lien avec la stabilité non-linéaire, et en fait la stabilité orbitale puisque ce sont des problèmes invariants par translation, est loin d’être clair. Car on ne peut espérer qu’une stabilité spectrale neutre, étant donné que la structure hamiltonnienne exclut l’existence d’un trou spectral, et ce même en faisant abstraction de la valeur propre nulle, liée à l’invariance par translation. D’autres méthodes pour étudier la stabilité des ondes progressives périodiques consistent à tirer parti de la structure sous-jacente. C’est naturellement le cas de l’approche variationnelle. Celle-ci consiste à utiliser le hamiltonnien, ou plus précisément une fonctionnelle modifiée pour tenir compte des autres quantités conservées, comme fonction de Lyapunov. Lorsqu’elle s’applique, cette méthode est très efficace et donne directement accès à la stabilité orbitale. Une troisième voie est la théorie de la modulation, dont les fondements ont été posés par Whitham à l’orée des années 1970. L’objectif est ici de présenter quelques résultats récents, valant pour des équations et systèmes du type de l’équation Korteweg–de Vries, qui mettent en relation les approches spectrale, variationnelle et modulationnelle.
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, 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 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.
@article{JEDP_2013____A2_0,
author = {Benzoni-Gavage, Sylvie and Noble, Pascal and Rodrigues, L.~Miguel},
title = {Stability of periodic waves in Hamiltonian PDEs},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
year = {2013},
pages = {1-22},
doi = {10.5802/jedp.98},
language = {en},
url = {http://dml.mathdoc.fr/item/JEDP_2013____A2_0}
}
Benzoni-Gavage, Sylvie; Noble, Pascal; Rodrigues, L. Miguel. Stability of periodic waves in Hamiltonian PDEs. Journées équations aux dérivées partielles, (2013), pp. 1-22. doi : 10.5802/jedp.98. http://gdmltest.u-ga.fr/item/JEDP_2013____A2_0/
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