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.

Mastropietro, V., & Procesi, M. (2006). Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 5(1), 1-28 [10.3934/cpaa.2006.5.1].

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

MASTROPIETRO, VIERI;
2006

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.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/07 - Fisica Matematica
English
Con Impact Factor ISI
Diophantine and irrationality conditions; Dirichlet boundary conditions; Lindstedt series method; Nonlinear wave equation; Periodic solutions; Perturbation theory; Tree formalism
arXiv:math/0505283v1
Mastropietro, V., & Procesi, M. (2006). Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 5(1), 1-28 [10.3934/cpaa.2006.5.1].
Mastropietro, V; Procesi, M
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/34434
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