EL, G.A. ... et al., 2009. Two-dimensional supersonic nonlinear Schrödinger flow past an extended obstacle. Physical Review (E), 80 (4), 046317.
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 suffciently 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.
A.M.K. thanks the
Royal Society for the financial support of his visit to Loughborough
University and RFBR Grant No. 09-02-00499 for
partial support. V.V.K. thanks the London Mathematical Society
for partial support of his visit to Loughborough University.