Tiziano Penati

Continuous approximation of breathers in one- and two-dimensional DNLS lattices

In this paper we construct and approximate breathers in the DNLS model starting from the continuous limit: such periodic solutions are obtained as perturbations of the ground state of the NLS model in , with n = 1, 2. In both the dimensions we recover the Sievers–Takeno and the Page (P) modes; furthermore, in also the two hybrid (H) modes are constructed. The proof is based on the interpolation of the lattice using the finite element method (FEM).

Boundary effects on the dynamics of chains of coupled oscillators

We study the dynamics of a chain of coupled particles subjected to a restoring force (Klein–Gordon lattice) in the cases of either periodic or Dirichlet boundary conditions. Precisely, we prove that, when the initial data are of small amplitude and have a long wavelength, the main part of the solution is interpolated by a solution of the nonlinear Schrodinger equation, which in turn has the property that its Fourier coefficients decay exponentially. The first order correction to the solution has Fourier coefficients that decay exponentially in the periodic case, but only as a power in the Dirichlet case. In particular our result allows one to explain the numerical computations of the paper (Bambusi et al 2007 Phys. Lett. A).

Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrödinger equations

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 second method is based on a resonant normal form theorem. We show that although the two methods are different in the implementation, they produce equivalent results as the end product. We also discuss the applications of the discrete nonlinear Schrodinger equation in the context of existence and stability of breathers of the Klein–Gordon lattice.

Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein–Gordon lattices

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 model. The proof is based on a Lyapunov–Schmidt decomposition and continuum approximation techniques introduced in [Bambusi and Penati, Continuous approximation of ground states in DNLS lattices, Nonlinearity 23 (2010), pp. 143–157], actually using its main result as an important lemma.

An Extensive Adiabatic Invariant for the Klein–Gordon Model in the Thermodynamic Limit

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 β1/a for any large enough value of the inverse temperature β. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbative constructions and the subsequent estimates are consistent with the extensive nature of the system.

Breathers and Q-Breathers: Two Sides of the Same Coin

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 Schrodinger models, with Dirichlet boundary conditions. Within such families, depending on the parameters, both real space localization (breathers) and Fourier space localization (Q-breathers) are present. For the former type of solutions, convergence to the ground state of the focusing infinite chain is also proved; for the latter, a description of the localization properties is given, and some numerical results on the difference between the focusing and defocusing cases are explained. The proofs are based on continuation tools, ideas from the finite element methods, and techniques of convergence of variational problems.

An extensive resonant normal form for an arbitrary large Klein–Gordon model

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 normal form is a generalized discrete nonlinear Schrödinger model, characterized by all-to-all sites coupling with exponentially decaying strength.

On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice

Abstract We consider a one-dimensional discrete nonlinear Schrodinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to four-site vortex solutions in the standard two-dimensional dNLS model (square lattice), of which this is a simpler variant. Due to the specific choice of lengths of the inter-site interactions, the vortex configurations considered present a degeneracy which causes the standard continuation techniques to be non-applicable. In the present one-dimensional case, the existence of a conserved quantity for the soliton profile (the so-called density current), together with a perturbative construction, leads to the nonexistence of any phase-shift discrete soliton which is at least C 2 with respect to the small coupling ϵ , in the limit of vanishing ϵ . If we assume the solution to be only C 0 in the same limit of ϵ , nonexistence is instead proved by studying the bifurcation equation of a Lyapunov–Schmidt reduction, expanded to suitably high orders. Specifically, we produce a nonexistence criterion whose efficiency we reveal in the cases of partial and full degeneracy of approximate solutions obtained via a leading order expansion.

On the continuation of degenerate periodic orbits via normal form: full dimensional resonant tori

Abstract 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 constructive normal form scheme that allows to identify and approximate the periodic orbits which continue to exist after the breaking of the resonant torus. A specific feature of our algorithm consists in the possibility of dealing with degenerate periodic orbits. Besides, under suitable hypothesis on the spectrum of the approximate periodic orbit, we obtain information on the linear stability of the periodic orbits feasible of continuation. A pedagogical example involving few degrees of freedom, but connected to the classical topic of discrete solitons in dNLS-lattices, is also provided.

Generalized Discrete Nonlinear Schrödinger as a Normal Form at the Thermodynamic Limit for the Klein–Gordon Chain

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., for a model of a crystal the number of particles should be of the order of the Avogadro number) with non vanishing energy per particle (which corresponds to a non zero temperature in the physical model).