Energy self-localization and gap local pulses in a two-dimensional nonlinear lattice

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Abstract
We study the formation of localized states, mediated by modulational instability, on a two-dimensional lattice with nonlinear coupling between nearest particles and a periodic nonlinear substrate potential. Such a discrete system can model molecules adsorbed on a substrate crystal surface, for example. The basic equations of the motion governing the dynamics of the lattice are derived from the model Hamiltonian. In the low-amplitude approximation and semidiscrete limit these equations can be approximated by a two-dimensional nonlinear Schrödinger equation. The modulational instability conditions are calculated; they inform us about the selection mechanism of the wave vectors and growth rate of the instabilities taking place both in the longitudinal and transverse directions. The dynamics of the lattice is then investigated by means of numerical simulations; due to modulational instability an initial steady state that consists of a plane wave with low amplitude modulated by very weak noise, evolves into an oscillating localized state, inhomogeneously distributed on the lattice. These nonlinear localized modes, which move slowly, present the remarkable properties of gap modes. Their amplitude is large and they pulsate at a low frequency that lies inside the lower linear gap of the lattice.