Two-dimensional supersonic nonlinear Schrödinger flow past an extended obstacle

Abstract

Supersonic flow of a superfluid past a slender impenetrable macroscopic obstacle is studied in the framework of the two-dimensional defocusing nonlinear Schrödinger (NLS) equation. This problem is of fundamental importance as a dispersive analogue of the corresponding classical gas-dynamics problem. Assuming the oncoming flow speed sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x-coordinate and the piston motion curve is defined by the spatial body profile. Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half-planes. These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW. We propose an extension of the traditional modulation description of DSWs to include the linear "ship wave" pattern forming outside the nonlinear modulation region of the front DSW. Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.

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

“…which agrees with the asymptotic representation of a highly supersonic oblique GP soliton in [17]. Without loss of generality we consider the waves in the upper half-plane (a > 0), then it follows from (44) that 0 ≤ λ ≤ 1.…”

“…The obtained solutions provide a basis for further studies connected with the description of dispersive shock waves observed in recent experiments [2] of the flow of a BEC past obstacles as well as in numerical simulations [17]. Some straightforward implications about the characteristic features of the wave patterns arising in the flow of a BEC past an obstacle have been made from the expressions for the slope a in the obtained asymptotic reductions of the full periodic solution.…”

“…As was noted above, the general formula (49) describes linear waves of an arbitrary wavelength and it can be mapped to the tails of the general soliton solution (17).…”

“…The stability of the oblique dark solitons (17) for M > 1 was established in [17] numerically. It is important now to investigate the behaviour of the obtained periodic solution for the parameter values corresponding to some physically interesting asymptotic reductions of the system (5).…”

“…As was indicated in [17], if we consider a flow of BEC corresponding to small deviations from a uniform and homogeneous supersonic flow with n = 1, u = M, v = 0 (M > 1), and make asymptotic expansions…”