Two-dimensional periodic waves in a supersonic flow of a Bose-Einstein condensate

Stationary periodic solutions of the two-dimensional Gross-Pitaevskii equation are obtained and analyzed for different parameter values in the context of the problem of a supersonic flow of a Bose-Einstein condensate past an obstacle. The asymptotic connections with the corresponding periodic solutions of the Korteweg-de Vries and nonlinear Schr\"odinger equations are studied and typical spatial wave distributions are discussed.