2005/05/13 by Vieri Mastropietro, Michela Procesi, Mastropietro, Vieri +1
Mathematics · Physics and Astronomy · #35B10 (primary) #35B32 #35L70 #47H15 (secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Photonic Systems #Numerical methods for differential equations #Quantum chaos and dynamical systems #math.AP #math.FA #msc:35B10 #msc:35B32 #msc:35L70 #msc:47H15
paper · pdf · doi:10.48550/arxiv.math/0505283
29 pages 6 figures
arxiv created 2005/05/13 · openalex publication_date 2005/05/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Dirichlet 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 means of a perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mechanics. The periodic solutions are not analytic but defined only in a Cantor set, and resummation techniques of divergent powers series are used in order to control the small divisors problem.