In this article we prove the existence of a new family of periodic solutions for discrete, nonlinear Schrödinger equations subject to spatially localized driving and damping and we show numerically that they provide a more accurate approximation to metastable states in these systems than previous proposals. We also study the stability properties of these solutions and show that they fit well with a previously proposed mechanism for the emergence and persistence of metastable behavior.

In this article we prove the existence of a new family of periodic solutions for discrete, nonlinear Schrödinger equations subject to spatially localized driving and damping. They provide an alternate description of the metastable behavior in such lattice systems which agrees with previous predictions for the evolution of metastable states while providing more accurate approximations to these states. We analyze the stability of these breathers, finding a very small positive eigenvalue whose eigenvector lies almost tangent to the surface of the cylinder formed by the family of breathers. This causes solutions to slide along the cylinder without leaving its neighborhood for very long times.