Faraday instability and Faraday patterns in a superfluid Fermi gas

With the consideration of the coupling between the transverse width and the longitudinal density, the parametric excitations related to Faraday waves in a cigar-shaped superfluid Fermi gas are studied. A Mathieu equation is obtained, and it is demonstrated firstly that the excited actual 3D Faraday pattern is the combination of the longitudinal Faraday density wave and the corresponding transverse width fluctuation in the longitudinal direction. The Faraday instability growth index and the kinematic equations of the Faraday density wave and the width fluctuation along the Bose-Einstein condensate (BEC)-Bardeen-Cooper-Schrieffer (BCS) crossover are also given for the first time. It is found that the 3D Faraday pattern presents quite different behaviours (such as the excitations and the motions) when the system crosses from the BEC side to the BCS side. The coupling not only plays an important role in the parametric excitation, but also determines the dominant wavelength of the spatial structure. Along the crossover, the coupling effects are more significant in the BCS side. The final numerical investigation verifies these results and gives a detailed study of the parametric excitations (i.e. Faraday instability) and the 3D pattern formation.