Parametrically Amplified 2-Dimensional Solitary Waves

Abstract

In this work, the existence of stable localized structures in extended dissipative 2D systems is analyzed. A model problem for a system where a parametric instability occurs is used. The PFDNLS equation which describes the complex amplitude of the parametric instability waves in a weakly non-linear and weakly dispersive system is numerically integrated. The numerical computations show that in the parameter space (forcing amplitude — detuning) a region exists where localized steady solutions are stable in the form ofaxisymetric pulses. This region of pulse stability occurs when uniform waves of NLS are Benjamin-Feir unstable (“focussing” case) and when the bifurcation is of sub-critical nature. Such localized structures have also been observed in Faraday instability experiments. Several bifurcations paths of these steady pulses have been studied and different at tractors have been found (oscillating pulses, steady multi-pulse, chaotic multi-pulse).