Gap and out-gap solitons in modulated systems of finite length: exact solutions in the slowly varying envelope limit

Abstract

We discuss nonlinear excitations in finite-size one-dimensional modulated systems. Considering a binary modulated discrete nonlinear Schrödinger chain of large but finite length with periodic boundary conditions, we obtain exact elliptic-function solutions corresponding to stationary excitations in the slowly varying envelope limit. From these solutions, we analyze how the transformation between (localized) gap and (delocalized) out-gap solitons manifests itself in a system of finite length. The analogue of a localized gap soliton appears through a bifurcation at a critical point, so that gap soliton analogues exist only for chains longer than a critical value, which scales inversely proportional to the modulation depth. The total norm of these gap–out-gap states is found to be a monotonic function of the frequency, always inside a ‘nonlinear gap’ with edges defined by the main nonlinear modes which approach the linear spectrum gap boundaries in the small-amplitude limit. The transformation from a gap to an out-gap state is associated with a particular frequency, close to the lower boundary of the linear gap; at this point the elliptic functions become trigonometric, corresponding to a finite-size analogue of an algebraic soliton. We compare the scenario with earlier results obtained numerically for purely discrete chains with few degrees of freedom.