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

We study the externally driven damped nonlinear Schr\"odinger 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 {γ}. For generic values of h and {γ} there are two coexisting solitons, one of which (ψ₊) is always unstable. The bifurcation diagram of the second soliton (ψ_-) depends on the dissipation coefficient: if {γ}γcr, the ψ_- is stable for small h and loses its stability via a Hopf bifurcation as h is increased; if {γ}>γcr, the ψ_- 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. {} 1996 The American Physical Society.