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

TL;DR

This paper proves the existence of small amplitude, time-periodic solutions for fully nonlinear Benjamin-Ono type equations, demonstrating multimodal solutions with multiple traveling waves in a reversible Hamiltonian system with perturbations.

Contribution

It introduces a novel method to handle the linearized operator with nonconstant coefficients and high-order derivatives, extending techniques for nonlinear pseudo-PDEs of Benjamin-Ono type.

Findings

Existence of time-periodic small amplitude solutions in Sobolev spaces.

Solutions are superpositions of multiple traveling waves with different velocities.

Frequency set has asymptotically full measure as amplitude approaches zero.

Abstract

We prove the existence of time-periodic, small amplitude solutions of autonomous quasilinear 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 nonconstant 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. (Version 2: small change in Section 2).