arXiv subjects

Michela Procesi

Publications and source records attributed to Michela Procesi.

41 records · Page 3Linked to original sources

arXiv2007

Periodic solutions for the Schroedinger equation with nonlocal smoothing nonlinearities in higher dimension

We consider the nonlinear Schroedinger equation in higher dimension with Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in solving a "small divisor problem" which we do by using a renormalisation group approach.

math.AP↗

arXiv2005

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

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.

math.FA↗

arXiv2005

Quasi-periodic solutions of completely resonant forced wave equations

We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.

math.FA↗

arXiv2004

Conservation of resonant periodic solutions for the one-dimensional nonlinear Schroedinger equation

We consider the one-dimensional nonlinear Schrödinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the zero-mass term). We investigate conservation of small amplitude periodic-solutions for a large set measure of frequencies. In particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes, provided the corresponding frequencies are large enough and close enough to each other (wave packets with large wave number).

math.DS↗

arXiv2004

Periodic solutions for completely resonant nonlinear wave equations

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=ϕ(u)$, with $ϕ(u)=Φu^{3} + O(u^{5})$ odd and analytic, $Φ\neq0$, and we construct small amplitude periodic solutions with frequency $ø$ for a large Lebesgue measure set of $ø$ close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and nonresonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect the nonlinear wave equations $u_{xx}-u_{tt}+ M u = ϕ(u)$, $M\neq0$, is that not only the P equation but also the Q equation is infinite-dimensional

math.DS↗