Nonlinear Periodic Systems: Bands and Localization

We consider the dynamics of nonlinear mono-coupled periodic media. When coupling dominates over nonlinearity near-field standing waves and spatially extended traveling waves exist, inside stop and pass bands, respectively, of the nonlinear system. Nonlinear standing waves are analytically studied using a nonlinear normal mode formulation, whereas nonlinear traveling waves are analyzed by the method of multiple scales. When the nonlinear effects are of the same order with the coupling ones a completely different picture emerges, since nonlinear resonance interactions are unavoidable. As a result, infinite families of strongly and weakly localized nonlinear standing waves appear with frequencies lying in pass or stop bands of the corresponding linear periodic medium. Moreover, in the limit of weak coupling these solutions develop sensitive dependence on initial conditions, and the possibility of spatial chaos in the system exists. Some additional results on chaotic dynamics in linear periodic media with strongly nonlinear disorders are reviewed.