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We investigate pattern formation in two-dimensional Bose-Einstein condensates(BECs) caused by temporal periodic modulation of the interatomic interaction.Temporal modulation of the interaction causes the so-called Faraday instabilityin the condensate, which we show generically leads to a stable square griddensity pattern. We take the amplitudes in each of the two directions spanningthe two-dimensional density pattern as order parameters in pattern formationand derive a set of simultaneous time evolution equations for those orderparameters from the Gross--Pitaevskii (GP) equation with a time-periodicinteraction. We identify the fixed points of the time evolution and show bystability analysis that the inhomogeneous density exhibits a square gridpattern as a stable fixed point.

Square Pattern Formation as Stable Fixed Point in Driven Two-Dimensional Bose-Einstein Condensates