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**11 - 19**of**19**### + LINDSTEDT SERIES FOR PERIODIC SOLUTIONS OF BEAM EQUATIONS WITH QUADRATIC AND VELOCITY DEPENDENT NONLINEARITIES

"... 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 m ..."

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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.

### unknown title

, 2004

"... Bifurcation of free vibrations for completely resonant wave equations ..."

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### Forced

, 2004

"... vibrations of wave equations with non-monotone nonlinearities ..."

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### OF BEAM EQUATIONS WITH QUADRATIC AND VELOCITY DEPENDENT NONLINEARITIES

, 2005

"... 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 Dirichlet boundary conditions. Such nonlinear PDE can be regarded as a simple mod ..."

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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 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.

### QUASI-PERIODIC SOLUTIONS OF THE EQUATION vtt − vxx + v 3 = f(v)

, 2005

"... Abstract. We consider 1D completely resonant nonlinear wave equations of the type vtt − vxx = −v 3 + O(v 4) with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutio ..."

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Abstract. We consider 1D completely resonant nonlinear wave equations of the type vtt − vxx = −v 3 + O(v 4) with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutions turn out to be, at the first order, the superposition of a traveling wave and a modulation of long period, depending only on time. 1.

### ISBN 978-80-227-2624-5 NONLINEAR OSCILLATIONS OF COMPLETELY RESONANT WAVE EQUATIONS∗

"... Abstract. We present recent existence and multiplicity results of small amplitude periodic solutions of completely resonant nonlinear wave equations with frequencies ω belonging to a Cantor-like set of asymptotically full measure. The proofs rely on a suitable Lyapunov-Schmidt decomposition, a varia ..."

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Abstract. We present recent existence and multiplicity results of small amplitude periodic solutions of completely resonant nonlinear wave equations with frequencies ω belonging to a Cantor-like set of asymptotically full measure. The proofs rely on a suitable Lyapunov-Schmidt decomposition, a variant of the Nash-Moser Implicit Function Theorem and Variational Methods.