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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

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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