Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities

V. Mastropietro; Michela Procesi

doi:10.3934/cpaa.2006.5.1

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2006, Volume 5, Issue 1: 1-28. Doi: 10.3934/cpaa.2006.5.1

This issue Previous Article Next Article Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions

Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities

V. Mastropietro^1, and
Michela Procesi^2,

1.
Dipartimento di Matematica, Università di Roma "Tor Vergata", Roma, I-00133
2.
Dipartimento di Matematica, Università di Roma Tre, Roma, I-00146

Received: March 31, 2005

Revised: October 31, 2005

Published: February 2006

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Abstract

We prove the existence of small amplitude periodic solutions, for a large Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak quadratic and velocity dependent nonlinearity and with Dirichelet boundary conditions. Such nonlinear PDE can be regarded as a simple model describing oscillations of flexible structures like suspension bridges in presence of an uniform wind flow. The periodic solutions are explicitly constructed by a convergent perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mechanics. The periodic solutions are defined only in a Cantor set, and resummation techniques of divergent powers series are used in order to control the small divisors problem.

Keywords:

Nonlinear wave equation,
periodic solutions,
Lindstedt series method,
tree formalism,
perturbation theory,
Diophantine and irrationality conditions,
Dirichlet boundary conditions.

Mathematics Subject Classification: Primary 35B10, 35B32, 35L70, 47H15, 47N20, 58F27, 58F39.

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