Breathers in a discrete nonlinear Schrödinger-type model: Exact stability results

Following our earlier work [Phys. Rev. Lett. 84, 3570 (2000)] we present an exact linear stability analysis of one-site monochromatic breathers in a piecewise smooth discrete nonlinear Schrödinger-type model. Destabilization of the breather occurs by virtue of a growth rate becoming positive as a stability border is crossed, while above a critical spatial decay rate (lambda(c)) the breather is found to be intrinsically unstable. The model admits of other exact breather solutions, including multisite monochromatic breathers for which the profile variable (phi(n)) crosses a relevant threshold at more than one site. In particular, we consider exact two-site breather solutions with phase difference delta between the two sites above threshold, and present stability results for delta=pi (antiphase breather; the in-phase breather with delta=0 happens to be intrinsically unstable). We obtain a band of extended eigenmodes, together with a pair of localized symmetric modes and another pair of localized antisymmetric ones. The frequencies of the localized modes vary as the parameters characterizing the breather are made to vary, and destabilization occurs through the Krein collision of a quartet of growth rates, leading to temporal growth of a pair of symmetric eigenmodes of nonzero frequency. We clarify the limit N--> infinity (N is the gap length between the sites above threshold) when the two-site breather reduces to a pair of decoupled one-site breathers. The model offers the possibility of obtaining spatially random vortex-type breathers.