# Simone PaleariUniversity of Milan | UNIMI · Department of Mathematics

Simone Paleari

PhD

## About

40

Publications

2,324

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

492

Citations

Introduction

Additional affiliations

May 2006 - January 2015

February 2006 - April 2006

January 2004 - December 2005

## Publications

Publications (40)

We consider the problem of the continuation with respect to a small parameter ɛ of spatially localized and time periodic solutions in 1-dimensional dNLS lattices, where ɛ represents the strength of the interaction among the sites on the lattice. Specifically, we consider different dNLS models and apply a recently developed normal form algorithm in...

We consider the problem of the continuation with respect to a small parameter $\epsilon$ of spatially localised and time periodic solutions in 1-dimensional dNLS lattices, where $\epsilon$ represents the strength of the interaction among the sites on the lattice. Specifically, we consider different dNLS models and apply a recently developed normal...

We present an extension of a classical result of Poincaré (1892) about continuation of periodic orbits and breaking of completely resonant tori in a class of nearly integrable Hamiltonian systems, which covers most Hamiltonian Lattice models. The result is based on the fixed point method of the period map and exploits a standard perturbation expans...

We consider a discrete Klein–Gordon (dKG) equation on Open image in new window in the limit of the discrete nonlinear Schrödinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling strength 𝜖. By using the classical Lyapunov–Schmidt method, we show existence and linear stability of the KG br...

We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results (presented in CNSNS, 61:198-224, 2018) for full dimensional resonant tori to lower dimensional ones. We develop a const...

We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results (presented in CNSNS, 61:198-224, 2018) for full dimensional resonant tori to lower dimensional ones. We develop a const...

Hamiltonian lattice dynamics is a very active and relevant field of research. In this Special Issue, by means of some recent results by leading experts in the field, we tried to illustrate how broad and rich it can be, and how it can be seen as excellent playground for Mathematics in Engineering.

In this work, we study the existence of low amplitude four-site phase-shift multibreathers for small values of the coupling $\epsilon$ in Klein-Gordon (KG) chains with interactions longer than the classical nearest-neighbour ones. In the proper parameter regimes, the considered lattices bear connections to models beyond one spatial dimension, namel...

In this work, we study the existence of, low amplitude, phase-shift multibreathers for small values of the linear coupling in Klein–Gordon chains with interactions beyond the classical nearest-neighbor (NN) ones. In the proper parameter regimes, the considered lattices bear connections to models beyond one spatial dimension, namely the so-called zi...

In this paper we consider a discrete Klein-Gordon (dKG) equation on $\ZZ^d$ in the limit of the discrete nonlinear Schrodinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling strength $\eps$. By using the classical Lyapunov-Schmidt method, we show existence and linear stability of the KG...

We consider a one-dimensional discrete nonlinear Schroedinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to four-sites vortex solutions in the standard two-dimensional dNLS model (square lattice), of which this is a simpler var...

In this work, we study the existence of low amplitude four-site phase-shift multibreathers for small values of the coupling $\epsilon$ in Klein-Gordon (KG) chains with interactions longer than the classical nearest-neighbour ones. In the proper parameter regimes, the considered lattices bear connections to models beyond one spatial dimension, namel...

In the limit of small couplings in the nearest neighbor interaction, and
small total energy, we consider a resonant normal form for a finite but
arbitrarily large mixed Fermi-Pasta-Ulam Klein-Gordon chain, i.e. in the
interaction part a nonlinear term is also allowed, with periodic boundary
conditions. The normal form turns out to be a generalized...

Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are
approximated by equations of the discrete nonlinear Schrodinger type. We show
how to justify this approximation by two methods, which have been very popular
in the recent literature. The first method relies on a priori energy estimates
and multi-scale decompositions. The se...

We construct an extensive adiabatic invariant for a Klein–Gordon chain in the thermodynamic limit. In particular, given a fixed and sufficiently small value of the coupling constant a, the evolution of the adiabatic invariant is controlled up to time scaling as β
1/a
for any large enough value of the inverse temperature β. The time scale becomes a...

A still open challenge in Hamiltonian dynamics is the development of a perturbation theory for Hamiltonian systems with an arbitrarily large number of degrees of freedom and, in particular, in the thermodynamic limit. Indeed, motivated by the problems arising in the foundations of Statistical Mechanics, it is relevant to consider large systems (e.g...

A still open challenge in Hamiltonian dynamics is the development of a perturbation theory for Hamiltonian systems with an arbitrarily large number of degrees of freedom and, in particular, in the thermodynamic limit. Indeed, motivated by the problems arising in the foundations of Statistical Mechanics, it is relevant to consider large systems (e.g...

We consider a finite but arbitrarily large Klein–Gordon chain, with periodic boundary conditions. In the limit of small couplings in the nearest neighbor interaction, and small (total or specific) energy, a high order resonant normal form is constructed with estimates uniform in the number of degrees of freedom. In particular, the first order norma...

We consider a system in which some high frequency harmonic oscillators are
coupled with a slow system. We prove that up to very long times the energy of
the high frequency system changes only by a small amount. The result we obtain
is completely independent of the resonance relations among the frequencies of
the fast system. More in detail, denote...

We look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit. Considering the quadratic part of the Klein-Gordon Hamiltonian, by a linear change of variables we transform it into a sum of two parts in involution. At variance with the usual method of introducing normal modes, our constructive procedure allows us to explo...

We reconsider the phenomenon of localization of energy in low frequency modes in the FPU system, exploiting the resonances in the lower part of the spectrum. Using the resonant normal form of Birkhoff we construct some candidates of approximate first integrals which we put in correspondence to packets of low frequency modes. By numerical calculatio...

We construct, and approximate from the continuum, two-parameter families of time periodic, small amplitude, localized solutions, for both the focusing and defocusing finite discrete nonlinear Schrödinger models, with Dirichlet boundary conditions. Within such families, depending on the parameters, both real space localization (breathers) and Fourie...

We construct small amplitude breathers in one-dimensional (1D) and two-dimensional (2D) Klein-Gordon (KG) infinite lattices. We also show that the breathers are well-approximated by the ground state of the nonlinear Schrödinger equation. The result is obtained by exploiting the relation between the KG lattice and the discrete nonlinear Schrödinger...

In this paper we give a statistical description of the phase space of a Fermi–Pasta–Ulam chain using the Fast Lyapunov Indicator,
looking for properties valid in the thermodynamic limit.

The well-known Fermi-Pasta-Ulam (FPU) phenomenon (lack of attainment of equipartition of the mode energies at low energies for some exceptional initial data) suggests that the FPU model does not have the mixing property at low energies. We give numerical indications that this is actually the case. This we show by computing orbits for sets of initia...

We show the relevance of the dispersive analogue of the shock waves in the FPU dynamics. In particular we give strict numerical evidence that metastable states emerging from low frequency initial excitations are indeed constituted by dispersive shock waves travelling through the chain. Relevant characteristics of the metastable states, such as thei...

We discuss the use of the maximal Lyapunov characteristic number (LCN) as a stochasticity indicator in connection with the persistence of the Fermi-Pasta-Ulam (FPU) paradox in the thermodynamic limit. We show that the positiveness of the LCN does not imply that the dynamic is ergodic in statistical sense. On the other hand, our numerical exploratio...

In this paper we report and discuss some recent results obtained investigating with numerical methods the celebrated Fermi--Pasta--Ulam model, a chain of non-linearly coupled oscillators with identical masses. We are interested in the evolution towards equipartition when energy is initially given to one or a few modes. Using the spectral entropy as...

We investigate with numerical methods the celebrated Fermi–Pasta–Ulam model, a chain of non–linearly coupled oscillators with identical masses. We are interested in the evolution towards equipartition when energy is initially given to one or a few modes. In previous works we considered the initial energy being given on the lower part of the spectru...

We investigate with numerical methods the scaling of the relaxation time to equipartition in the celebrated Fermi–Pasta–Ulam model. Our numerical results strongly suggest that the time increases exponentially with an inverse power of the specific energy. Such a scaling appears to remain valid in the thermodynamic limit.

In this paper we consider the following nonlinear plate equations: u(tt) + DeltaDeltau + mu = psi(x, u), psi(x, u) = +/-u(3) + O(u(5)), psi(-x, -u) = -psi(x, u), with Navier boundary conditions in a n-dimensional cube, here psi is a C-infinity function, and m is a positive parameter. For this equation we construct some Cantor families of periodic o...

We present here some results recently obtained, concerning the existence of families of small amplitude periodic orbits for some nonlinear PDEs. We consider equations whose linear part is completely resonant, with an application to a nonlinear string equation; for such a system we also give a stability result of Nekhoroshev type for the fundamental...

We study small amplitude solutions of the nonlinear wave equation utt ¡ uxx = ˆ(u) u(0;t )=0= u (…;t) (0.1) with an analytic nonlinearity of the type ˆ(u )= § u 2 k ¡ 1 + O ( u 2 k ), k ‚2 . For this system we introduce a new method to compute the resonant normal form obtaining a simple formula for it. Then we specialize to the case k = 2 and flnd...

In this review we study the periodic behaviour of some nonlinear PDEs looked upon as infinite dimensional dynamical systems. We briefly illustrate the origins and the motivations of this kind of problems and some of the lines along with they have been tackled. We explain then our contribution, which consists in a suitable
combination of bifurcation...

In this paper we study existence of families of periodic orbits close to equilibria of Lagrangian PDE's. We begin by a short survey on the problem, then we illustrate a method recently introduced by the authors to construct small amplitude periodic solutions of some semilinear partial differential equations. As a difference with respect to standard...

We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial differential equation. To this end, we construct, using averaging methods, a suitable map from the configuration space to itself. We prove that to each nondegenerate zero of such a map there correspond...

By means of nonsmooth critical-point theory, we prove existence of infinitely many solutions (u^m ) ⊆ H_0^1 (Ω, R^N ) for a class of perturbed Z_2−symmetric elliptic systems .

. We study the nonlinear wave equation u tt Gamma c 2 uxx = /(u) u(0; t) = 0 = u(; t) (0.1) with an analytic nonlinearity of the type /(u) = Sigmau 3 + P k4 ff k u k . On each small--energy surface we consider a solution of the linearized system with initial datum having the profile of an elliptic sinus: we show that solutions starting close to the...