On the Stability of Standing Solitons in Faraday Resonance

Abstract

This paper investigates the basic stability properties of the standing soliton occurring in Faraday resonance. We start by constructing a complete picture of the linearized stability of the soliton solutions of the forced, damped, nonlinear Schrödinger (NLS) equation. The linear-stability analysis shows that a small region of the parameter space is stable. The stable region is bordered by regions in which the soliton is unstable to the continuous spectrum of the linear operator, unstable to discrete spectrum of the linear operator or unstable to both. We perform numerical simulations of the forced, damped, NLS equation to determine the evolution of the solution after the onset of instability. The simulations fully confirm the predictions of the linear theory in each of the various régimes. Lastly, broad experimental observations are carried out to investigate the evolution of the solitons in the laboratory. The observations qualitatively confirm all the predicted behavior for the evolution of the soliton in the stable and unstable parameter régimes when the amplitude of the forcing and the frequency detuning are small.