Dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms

Experiments on free expansion of Bose-Einstein condensate (BEC) have shown [1] that evolution of large and smooth distributions of BEC is described very well by hydrodynamic approximation [2] where dispersion and dissipation effects are neglected. At the same time, it is well known from classical compressible gas dynamics (see, e.g., [3]) that typical initial distributions of density and velocity can lead to wave breaking phenomenon when formal solution of hydrodynamical equations becomes multivalued. It means that near the wave breaking point one cannot neglect dispersion and/or dissipation effects which prevent formation of a multivalued region of a solution. If the dissipation effects dominate the dispersion ones, then the multivalued region is replaced by the classical shock, i.e., narrow layer with strong dissipation within, which separates smooth regions with different values of density, fluid velocity and other physical parameters. This situation was studied in classical gas dynamics and found many practical applications. If, however, the dispersion effects dominate dissipation ones, then the region of strong oscillations is generated in the vicinity of the wave breaking point [4,5]. Observation of dark solitons in BEC [6–8] shows that the main role in dynamics of BEC is played by dispersion and nonlinear effects taken into account by the standard Gross-Pitaevskii (GP) equation [9], and dissipation effects are relatively small and can be considered as perturbation. Hence, there are initial distributions of BEC which can lead to formation of dissipationless shock waves. Here we shall consider such a possibility. The starting point of our consideration is the fact that the sound velocity in BEC is proportional to the square root from its density (see, e.g., [9] and references therein). Thus, if we create inhomogeneous BEC with high density hump (with density ,r1) in the center of lower density distribution (with density ,r0), and after that release this central part of BEC, then the high density hump will tend to expand with velocity ,Îr1 greater than the sound velocity ,Îr0 of propagation of disturbance in lower density BEC. As a result, wave breaking and formation of dissipationless shock wave can occur in this case. Note that initial distributions of this type were realized in experiment [10] on measurement of sound velocity in BEC and in the recent experiment [11]. In [10] the hump’s density r1 was too small to generate shocks (see below). In experiment [11] generation of shock oscillations was apparently observed. The theory of dissipationless shock waves in media described by a one-dimensional (1D) nonlinear Schrödinger (NLS) equation was developed in [12,13]. Since the GP equation in some cases can be reduced to the 1D NLS equation, this theory can be applied to the description of dissipationless shock waves in BEC. We consider BEC confined in a disk-shaped trap with the axial frequency vz much greater than the transverse one vx =vy =v'. We suppose that the lower density disk-shaped BEC is confined by magnetic trap and density distribution has standard Thomas-Fermi (TF) parabolic form. Let an additional potential be applied to BEC in the central part of TF profile which leads to narrow parabolic hump in the density distribution. After the central potential is switched off, the hump starts to expand against wide lower density TF profile leading to generation of oscillations in the transition region between high and low density condensates. Let the density of atoms in the central part of BEC be of order of magnitude n0 and satisfy the condition n0asaz !1, where as.0 is the s-wave scattering length and az= s" /mvzd is the amplitude of quantum oscillations in the axial trap. Then the condensate wave function c can be factorized as c=fszdCsx ,yd, where fszd=paz exps−z2 /2az d is the ground state wave function of axial motion, and Csx ,y , td satisfies the reduced 2D GP equation