Continuation of normal modes in finite NLS lattices

We study the continuation of breather solutions of the discrete NLS equation as the intersite coupling parameter is varied. Considering the case of a finite one-dimensional lattice of N sites, we show the existence of N branches of breathers that persist for arbitrary coupling, thus connecting normal modes of the linear system to breathers of the uncoupled, anticontinuous limit system. The proof is based on global bifurcation theory, applied to the continuation from the weakly nonlinear regime. As the coupling parameter varies these solutions generally change their stability, and we detect parameter regions where trajectories starting near unstable breathers appear to reach equipartition of power.