Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D

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April 2005, 13(3): 541-552. doi: 10.3934/dcds.2005.13.541

Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D

1.	SISSA- Via Beirut 2-4- 34014 Trieste, Italy

Received October 2004 Revised January 2005 Published May 2005

Abstract
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We provide quasi-periodic solutions with two frequencies $\omega\in \mathbb R^2$ for a class of completely resonant non-linear wave equations in one and two spatial dimensions and with periodic boundary conditions. This is the first existence result for quasi-periodic solutions in the completely resonant case. The main idea is to work in an appropriate invariant subspace, in order to simplify the bifurcation equation. The frequencies, close to that of the linear system, belong to an uncountable Cantor set of measure zero where no small divisor problem arises.

Keywords: infinite dimensional Hamiltonian systems, Lyapunov-Schmidt reduction., quasi-periodic solutions, Nonlinear wave equation.

Mathematics Subject Classification: 35L05, 37K50, 37C5.

Citation: Michela Procesi. Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 541-552. doi: 10.3934/dcds.2005.13.541

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