Variational treatment of Faraday waves in inhomogeneous Bose–Einstein condensates

Motivated by the recent experimental progress on the collective modes of a Bose–Einstein condensate whose atomic scattering length is tuned via Feshbach resonances, we analyze by variational means the dynamics of Faraday waves in trapped Bose–Einstein condensates. These waves can be excited by modulating periodically either the strength of the magnetic trap or the atomic scattering length. To study their dynamics, we develop a variational model that describes consistently both the bulk part of an inhomogeneous, low-density, cigar-shaped condensate and small-amplitude, small-wavelength Faraday waves. The main ansatz used in the variational treatment is tailored around a set of Gaussian envelopes and we show extensions for the high-density regime using a q -Gaussian function. Finally, we show explicitly that for drives of small amplitude, the two methods of obtaining Faraday waves are equivalent, and we discuss the existing experimental results.