Cantor families of periodic solutions for completely resonant nonlinear wave equations

Abstract

We prove the existence of small amplitude, (2π/ω)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency ω belonging to a Cantor-like set of asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finite dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity

KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

“…This characterization of the set of parameters which fulfill all the required Melnikov non-resonance conditions (at any step of the iteration) was first observed in [6], [5] in an analytic setting. Theorem 4.1 extends this property also in a differentiable setting.…”

Cantor families of periodic solutions of wave equations with C k nonlinearities

“…We assume the following nondegeneracy condition (of KAM type) which can be verified on several examples, see [4].…”

“…The main changes to be introduced to prove Theorem 1.1 with respect to the method of [4], regard the solution of the range equation through a differentiable Nash-Moser iterative scheme. This is done in sections 4 and 5, see remarks 4.1, 4.2, 4.3, 4.4.…”

“…The aim of this paper is to prove the bifurcation of "large" Cantor families of small amplitude periodic solutions for wave equations like In order to focus the attention on the main issue -namely the solution of the range equation via a differentiable Nash-Moser iteration scheme-we consider the completely resonant nonlinear wave equations of [4], without making any attempt to deal with nonlinearities which would require to solve differently the bifurcation equation.…”

“…The function u(δ) defines a C 1 Whitney extension of the family of periodic solutions of (1.2) constructed in the above theorem, see [4].…”