Localized excitations in two-dimensional Hamiltonian lattices

Abstract

We analyze the origin and features of localized excitations in a discrete two-dimensional Hamiltonian lattice. The lattice obeys discrete translational symmetry, and the localized excitations exist because of the presence of nonlinearities. We connect the presence of these excitations with the existence of local integrability of the original N degree of freedom system. On the basis of this explanation we make several predictions about the existence and stability of these excitations. This work is an extension of previously published results on vibrational localization in one-dimensional nonlinear Hamiltonian lattices (Phys.Rev.E.49(1994)836). Thus we confirm earlier suggestions about the generic property of Hamiltonian lattices to exhibit localized excitations independent on the dimensionality of the lattice. PACS number(s): 03.20.+i ; 63.20.Pw ; 63.20.Ry

Discrete breathers

“…if these perturbations are linearly stable, then we obtain bound states which are not merely a consequence of the existence of the DB solution itself. Such eigenstates were observed in many numerical studies [157], [46], [52], [87], [90], [88], [84], and especially studied systematically by Marin and Aubry in [140].…”

“…To see that we plot in Fig.10 a Poincare map of a perturbed DB of a two-dimensional lattice, where consecutive points are connected with straight lines. The attractor-like behaviour is seen, together with a seemingly remaining nonzero distance from the fixed point (periodic orbit) in the middle of the figure (for details see [84]). …”

“…For a complete discussion see Refs. [87], [90], [84]. We will finally discuss the possibility of wavelet analysis of breather-like objects [106].…”

Interaction of discrete breathers with electrons in nonlinear lattices

“…It is known that breather solutions exist, when essentially one (central) atom is performing periodic oscillations, with all the other atoms having exponentially small amplitudes [6], [8], [14]. Discrete breathers with such strong localization properties were found in numerical studies for onedimensional and two-dimensional lattices with moderate coupling and anharmonicity [4], [5]. Thus we assume the easiest form of the breather solution to be, that only one central particle is oscillating (with a frequency outside the linear phonon spectrum).…”

“…The concept of discrete breathers has been studied in detail in a number of publications [1], [2], [3], [4], [5], [6], [7], [8]. Discrete breathers are time-periodic solutions of a system of coupled classical degrees of freedom, typically arranged on a translationally invariant lattice.…”