Dynamics of the parametrically driven NLS solitons beyond the onset of theoscillatory instability

Solitary waves in conservative and near-conservative systems may become uns table due to a resonance of two internal oscillation modes. We study the pa rametrically driven, damped nonlinear Schrodinger equation, a prototype sys tem exhibiting this oscillatory instability An asymptotic multi-scale expan sion is used to derive a reduced amplitude equation describing the nonlinea r stage of the instability and supercritical dynamics of the soliton in the weakly dissipative case. We also derive the amplitude equation in the stro ngly dissipative case, when the bifurcation is of the Hopf type. The analys is of the reduced equations shows that in the undamped case the temporally periodic spatially localized structures are suppressed by the nonlinearity- induced radiation. In this case the unstable stationary soliton evolves eit her into a slowly decaying long-lived breather, or into a radiating soliton whose amplitude grows without bound. However, adding a small damping is su fficient to bring about a stably oscillating soliton of finite amplitude.