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 …