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

Abstract

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.