Existence and non-existence of breather solutions in damped and driven nonlinear lattices

Abstract

We investigate the existence of spatially localised solutions, in the form of discrete breathers, in general damped and driven nonlinear lattice systems of coupled oscillators. Conditions for the exponential decay of the difference between the maximal and minimal amplitudes of the oscillators are provided which proves that initial non-uniform spatial patterns representing breathers attain exponentially fast a spatially uniform state preventing the formation and/or preservation of any breather solution at all. Strikingly our results are generic in the sense that they hold for arbitrary dimension of the system, any attractive interaction, coupling strength and on-site potential and general driving fields. Furthermore, our rigorous quantitative results establish conditions under which discrete breathers in general damped and driven nonlinear lattices can exist at all and open the way for further research on the emergent dynamical scenarios, in particular features of pattern formation, localisation and synchronisation, in coupled cell networks.