Discrete breathers and solitons in metamaterials

Nonlinear localization is a process that may occur in weakly coupled nonlinear oscillators and leads to the formation of dynamically localized states in an otherwise translationally invariant lattice [1-3]. The main ingredients of nonlinear localization is discreteness, usually stemming from the weak interaction among the oscillators and nonlinearity, arising from the nonlinear nature of the oscillator forces. The dynamical localized states generated in this process are termed discrete breathers (DBs) or intrinsic localized modes (ILMs). These states are collective periodically oscillating modes of the lattice that, at the same time, are localized in a given location of the system. One basic criterion for the formation of DBs in infinite lattices is that their frequency and its sidebands should not coinside with the linearized spectrum of the oscillator lattice [4]. When the interparticle interaction exceeds a certain threshold, DBs become unstable and ultimately disappear. However, if the coupling becomes strong enough, it is possible in some cases to still form localized states that are very extended and have features of nontopological solitons or solitary waves [5]. In the present chapter we will address the generation of localization through nonlinearity both in the discrete, weakly interacting limit, as well in the continuous one where the nonlinear excitations are much larger than the lattice spacing. In all cases we will be focusing on metamaterials made typically of micron sized units that provide desired system electromagnetic properties.