Existence and stability chart for the ac-driven, damped nonlinear Schrodinger solitons

We study the externally driven damped nonlinear Schrodinger equation on an infinite line. The existence and stability chart for its soliton solution is constructed on the plane of two control parameters: the forcing amplitude h and the dissipation coefficient gamma. For generic values of h and gamma there are two coexisting solitons, one of which (psi(+)) is always unstable. The bifurcation diagram of the second soliton (psi(-)) depends on the dissipation coefficient: if gamma < gamma(cr), the psi(-) is stable for small h and loses its stability via a Hopf bifurcation as h is increased; if gamma > gamma(cr), the psi(-) is stable for all h. There are no ''stability windows'' in the unstable region. We show that the previously reported stability windows occur only when the equation is considered on a finite (and small) spatial interval.