Double parametric resonance for matter-wave solitons in a time-modulated trap

We analyze the motion of solitons in a self-attractive Bose-Einstein condensate, loaded into a quasi-onedimensional parabolic potential trap, which is subjected to time-periodic modulation with an amplitude « and frequency V. First, we apply the variational approximation, which gives rise to decoupled equations of motion for the center-of-mass coordinate of the soliton, jstd, and its width astd. The equation for jstd is the ordinary Mathieu equation sMEd sit is an exact equation that does not depend on the adopted ansatzd, the equation for astd being a nonlinear generalization of the ME. Both equations give rise to the same map of instability zones in the s«,Vd plane, generated by the parametric resonances sPRsd, if the instability is defined as the onset of growth of the amplitude of the parametrically driven oscillations. In this sense, the double PR is predicted. Direct simulations of the underlying Gross-Pitaevskii equation give rise to a qualitatively similar but quantitatively different stability map for oscillations of the soliton’s width astd. In the direct simulations, we identify the soliton dynamics as unstable if the instability sagain, realized as indefinite growth of the amplitude of oscillationsd can be detected during a time comparable with, or smaller than, the lifetime of the condensate stherefore accessible to experimental detectiond. Two-soliton configurations are also investigated. It is concluded that multiple collisions between solitons are elastic, and they do not affect the instability borders.