Polygonal patterns of Faraday water waves analogous to collective excitations in Bose–Einstein condensates*

Since the celebrated discovery of beautiful ``crispations" on a vibrating fluid layer, Faraday experiment has been an everlasting research topic owing to its rich spectrum of wave phenomena. Here we report the observation of polygonal Faraday patterns on the water surface held in vibrating containers with parabolic and other concave bases. These patterns manifest themselves as simple geometric figures of $l$-fold symmetry, ranging from ellipse ($l = 2$) to heptagon ($l = 7$), with wavelength much longer than the capillary length. Hence, they are intrinsically different from the previously studied patterns in vibrating drops or puddles and represent a peculiar variety of nonlinear shallow-water gravity waves or tidal waves in concave basins. What is of particular interest is their resemblance to the star-shaped collective excitations recently discovered in a driven Bose-Einstein condensate, not only sharing identical square-root scaling dispersion and pattern dynamics, but also possessing similar nonlinear features like hard-spring nonlinearity. Based on the close correspondence, we propose an analogue of the patterning dynamics between the classical and quantum fluid systems subject to confinement and argue that this analogue is mathematically valid in nonlinear regime.