Numerical solution of the non-linear Schrodinger equation using
  smoothed-particle hydrodynamics
  We formulate a smoothed-particle hydrodynamics numerical method,
traditionally used for the Euler equations for fluid dynamics in the context of
astrophysical simulations, to solve the non-linear Schrodinger equation in the
Madelung formulation. The probability density of the wavefunction is
discretized into moving particles, whose properties are smoothed by a kernel
function. The traditional fluid pressure is replaced by a quantum pressure
tensor, for which a novel, robust discretization is found. We demonstrate our
numerical method on a variety of numerical test problems involving the simple
harmonic oscillator, Bose-Einstein condensates, collapsing singularities, and
dark matter halos governed by the Gross-Pitaevskii-Poisson equation. Our method
is conservative, applicable to unbounded domains, and is automatically adaptive
in its resolution, making it well suited to study problems with collapsing
solutions.
