EXISTENCE AND STABILITY CHART FOR THE AC-DRIVEN, DAMPED NONLINEAR SCHRODINGER SOLITONS

We study the externally driven damped nonlinear Schrodinger equation o n an infinite line. The existence and stability chart for its soliton solution is constructed on the plane of two control parameters: the fo rcing amplitude h and the dissipation coefficient gamma. For generic v alues of h and gamma there are two coexisting solitons, one of which ( psi(+)) is always unstable. The bifurcation diagram of the second soli ton (psi(-)) depends on the dissipation coefficient: if gamma < gamma( cr), the psi(-) is stable for small h and loses its stability via a Ho pf bifurcation as h is increased; if gamma > gamma(cr), the psi(-) is stable for all h. There are no ''stability windows'' in the unstable r egion. We show that the previously reported stability windows occur on ly when the equation is considered on a finite (and small) spatial int erval.