Generation of Cherenkov waves in the flow of a Bose–Einstein condensate past an obstacle

Abstract

The theory of stationary linear wave patterns generated in a supersonic flow of a Bose–Einstein condensate past a point-like obstacle is developed. It is shown that they are located mainly outside the Mach cone corresponding to infinitely long wavelengths. The shape of wave crests and dependence of amplitude on coordinates far enough from the obstacle are calculated. The results are in good agreement with the results of numerical simulations. The theory gives a qualitative description of experiments with Bose–Einstein condensate flow past an obstacle after the condensate's release from a trap.

Two-dimensional supersonic nonlinear Schrödinger flow past an extended obstacle

“…11-13) and should be taken into account. A similar wave distribution was considered recently in [31,32,33] in connection with the Bogoliubov-Kelvin "ship waves" generated by a point-like obstacle placed in the supersonic BEC flow (see also in [26] the discussion of the experimentally observed patterns). An extended modulation solution describing the combined wave pattern including both the DSW and the linear "ship-wave" distribution will be constructed in the next section.…”

“…The lines of constant phase in this linear modulation solution determine the location of the small-amplitude wavecrests visible in numerical and physical experiments. Together with the DSW, they form a structure which eventually transforms into the universal Kelvin-Bogoliubov "ship wave" pattern [31,32]. The far-field asymptotic behaviour of our nonlinear modulation solution describes the distributions of the wave amplitude in this "ship wave" as a function of the wing profile.…”

“…This "slender body" problem is fundamentally important as a dispersive counterpart of the classical gas dynamics problem about the supersonic flow past a "wing" (see, for instance, [1,2]) and has an advantage of the possibility of full analytical treatment. In addition, the solution of this problem elucidates the macroscopic mechanisms of the generation of dark solitons and "ship waves" in BECs observed in the numerical and physical experiments [25,26,29,31,32,33,35]. Foreseeable direct physical applications could be connected with the BEC flows in atom-chip systems (see, e.g., [36,37] and references therein).…”

“…In the recent studies [29,31,32,33] of the supersonic BEC flow past localized obstacles two main distinct ingredients of the generated wave pattern have been identified and studied analytically and numerically: the so-called "ship waves" corresponding to the spatial Bogoliubov modes and generated outside the Mach cone, and oblique dark solitons generated inside the Mach cone and stretching behind the obstacle. An unexpected feature of these oblique dark solitons, established first numerically in [29], is their apparent stability, in striking contrast with the well established notion of the absolute "snake" instability of two-dimensional dark NLS solitons leading to their decay into vortexantivortex pairs [39,40,41], [42].…”

“…We note that in [29,31,32,33] the ship waves and oblique dark solitons were studied as separate independent wave structures generated by an idealized obstacle of small size placed in the BEC flow. At the same time, the process of the generation of these wave structures, as well as the connection of their parameters with the geometry and size of the physical obstacle remained beyond the scope of the cited studies.…”