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

Solitary waves in conservative and near-conservative systems may become unstable due to a resonance of two internal oscillation modes. We study the parametrically driven, damped nonlinear Schrödinger equation, a prototype system exhibiting this oscillatory instability. An asymptotic multi-scale expansion is used to derive a reduced amplitude equation describing the nonlinear stage of the instability and supercritical dynamics of the soliton in the weakly dissipative case. We also derive the amplitude equation in the strongly dissipative case, when the bifurcation is of the Hopf type. The analysis 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 either into a slowly decaying long-lived breather, or into a radiating soliton whose amplitude grows without bound. However, adding a small damping is sufficient to bring about a stably oscillating soliton of finite amplitude.