Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type

Baldi, Pietro

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 1, pp. 33-77.

Nous démontrons lʼexistence de solutions de petite amplitude, périodiques en temps, de versions quasi-linéaires ou complètement non linéaires de lʼéquation de Benjamin–Ono, dans le cas complètement résonnant. Le résultat est démontré dans lʼéchelle de Sobolev, pour des fréquences dans un ensemble de Cantor de mesure asymptotiquement pleine à lʼorigine. Nous considérons le cas général où lʼéquation peut-être vue comme un système Hamiltonien réversible perturbé par une partie réversible, mais qui nʼest pas Hamiltonienne, contenant des dérivées dʼordre maximal.Ces solutions sont, au premier ordre, obtenues par superposition dʼun nombre arbitrairement grand dʼondes se propageant à des vitesses différentes (solutions multimodales).Les principales difficultés du problème sont : la présence dʼun noyau de dimension infinie pour lʼéquation de bifurcation, et lʼoccurence de petits diviseurs dans la résolution de lʼéquation linéarisée, où les dérivées de plus haut degré ont des coefficients variables.Nous montrons que lʼopérateur linéarisé est essentiellement conjugué à un opérateur à coefficients constants, modulo un terme régularisant. La démonstration, basée sur des changements de variables et des conjugaisons avec des opérateurs pseudo-différentiels, est inspirée par la méthode utilisée par Iooss, Plotnikov et Toland (ARMA 2005) pour démontrer lʼexistence dʼondes de gravité stationnaires. La démonstration utilise également un schéma de Nash–Moser adapté à ce contexte, dans lʼéchelle des espaces de Sobolev, ainsi quʼune décomposition de Lyapunov–Schmidt.

We prove the existence of time-periodic, small amplitude solutions of autonomous quasi-linear or fully nonlinear completely resonant pseudo-PDEs of Benjamin–Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero.At the first order of amplitude, the solutions are the superposition of an arbitrarily large number of waves that travel with different velocities (multimodal solutions).The equation can be considered as a Hamiltonian, reversible system plus a non-Hamiltonian (but still reversible) perturbation that contains derivatives of the highest order.The main difficulties of the problem are: an infinite-dimensional bifurcation equation, and small divisors in the linearized operator, where also the highest order derivatives have non-constant coefficients.The main technical step of the proof is the reduction of the linearized operator to constant coefficients up to a regularizing rest, by means of changes of variables and conjugation with simple linear pseudo-differential operators, in the spirit of the method of Iooss, Plotnikov and Toland for standing water waves (ARMA 2005). Other ingredients are a suitable Nash–Moser iteration in Sobolev spaces, and Lyapunov–Schmidt decomposition.

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