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

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March 2006, 5(1): 1-28. doi: 10.3934/cpaa.2006.5.1

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, Italy

Received April 2005 Revised November 2005 Published December 2005

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

Mathematics Subject Classification: Primary 35B10, 35B32, 35L70, 47H15, 47N20, 58F27, 58F3.

Citation: V. Mastropietro, Michela Procesi. Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. Communications on Pure & Applied Analysis, 2006, 5 (1) : 1-28. doi: 10.3934/cpaa.2006.5.1

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