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Massimiliano Berti

Publications and source records attributed to Massimiliano Berti.

49 records · Page 3Linked to original sources
arXiv2010

Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential

We prove the existence of quasi-periodic solutions for Schroedinger equations with a multiplicative potential on T^d, d \geq 1, merely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are infinitely many times differentiable so are the solutions. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators ("Green functions") along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and "complexity" estimates.

math.AP↗
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↗
arXiv2003

A Birkhoff--Lewis Type Theorem for Some Hamiltonian Pdes

In this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schrödinger equation with smoothing nonlinearity.

math.DS↗
arXiv2003

Periodic orbits close to elliptic tori and applications to the three-body problem

We prove, under suitable non-resonance and non-degeneracy ``twist'' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the ``planets''. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.)

math.DS↗
arXiv2002

Periodic solutions of nonlinear wave equations with general nonlinearities

We prove the existence of small amplitude periodic solutions, with strongly irrational frequency $ \om $ close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For $ \om $ close to one we prove the existence of a large number $ N_\om $ of $ 2 π\slash \om $-periodic in time solutions $ u_1, ..., u_n, ..., u_N $: $ N_\om \to + \infty $ as $ \om \to 1 $. The minimal period of the $n$-th solution $u_n $ is proved to be $2 π\slash n \om $. The proofs are based on a Lyapunov-Schmidt reduction and variational arguments.

math.AP↗
arXiv2002

Drift in phase space: a new variational mechanism with optimal diffusion time

We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(μ)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $ T_d = O((1/ μ) \log (1/ μ))$ by a variational method which does not require the existence of ``transition chains of tori'' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d $ is optimal as a consequence of a general stability result derived from classical perturbation theory.

math.FA↗
arXiv2002

Optimal stability and instability results for a class of nearly integrable Hamiltonian systems

We consider a nearly integrable, non-isochronous, a-priori unstable Hamiltonian system with a (trigonometric polynomial) $O(μ)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $T_d = O((1/ μ) \log (1/ μ))$ by a variational method which does not require the existence of ``transition chains of tori'' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d$ is optimal as a consequence of a general stability result proved via classical perturbation theory.

math.DS↗
arXiv2001

Fast Arnold Diffusion in three time scale systems

We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the ``splitting determinant'' is exponentially small.

math.DS↗
arXiv2000

Arnold's Diffusion in nearly integrable isochronous Hamiltonian systems

We consider the problem of Arnold's diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also develop a new method for measuring the splitting of the separatrices. As an application we justify, for three time scales systems, that the splitting is correctly predicted by the Poincaré-Melnikov function.

math.DS↗
arXiv2000

Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium

We consider autonomous Lagrangian systems with two degrees of freedom, having an hyperbolic equilibrium of saddle-saddle type (that is the eingenvalues of the linearized system about the equilibrium are $\pm λ_1, \pm λ_2 $, $λ_1, λ_2 > 0$). We assume that $λ_1 > λ_2$ and that the system possesses two homoclinic orbits. Under a nondegeneracy assumption on the homoclinics and under suitable conditions on the geometric behaviour of these homoclinics near the equilibrium we prove, by variational methods, then they give rise to an infinite family of multibump homoclinic solutions and that the topological entropy at the zero energy level is positive. A method to deal also with homoclinics satisfying a weaker nondegeneracy condition is developed and it is applied, for simplicity, when $λ_1 \approx λ_2$. An application to a perturbation of a uncoupled system is also given.

math.DS↗