  In this article we study the one-dimensional, asymptotically linear,
non-linear Schr\"odinger equation (NLS). We show the existence of a global
smooth curve of standing waves for this problem, and we prove that these
standing waves are orbitally stable. As far as we know, this is the first
rigorous stability result for the asymptotically linear NLS. We also discuss an
application of our results to self-focusing waveguides with a saturable
refractive index.
