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;
2006-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.