Scalar Field Cosmologies Hidden Within the Nonlinear Schrodinger
  Equation
  The nonlinear, cubic Schrodinger (NLS) equation has numerous physical
applications, but in general is very difficult to solve. Nonetheless, under
certain circumstances parameters quantifying the width, momentum and energy of
the wavefunction evolve under a closed set of ordinary differential equations.
It is shown that for the case of the radial, two dimensional NLS equation, such
evolution equations may be mapped directly onto the cosmological Friedmann
equations for a spatially flat and isotropic universe sourced by a
self-interacting scalar field and a barotropic perfect fluid. Consequently,
analytical techniques that have been developed to study the dynamics of such
cosmological models may be applied to gain insight into aspects of nonlinear
quantum mechanics. In this paper, the Hamilton-Jacobi formalism of the
Friedmann equations, where the scalar field is viewed as the dynamical
variable, is developed within this context. Algorithms for finding exact
solutions are presented and the scaling solutions determined. A form-invariance
of the wavefunction evolution equations is identified. The analysis has direct
applications to anisotropic Bose-Einstein condensation. The Ermakov-Pinney
equation plays a central role in establishing the correspondence between the
quantum-mechanical and gravitational systems.
