  The large-scale expansion dynamics of quantum gases is a central tool for
ultracold gas experiments and poses a significant challenge for theory. In this
work we provide an exact reformulation of the Gross-Pitaevskii equation for the
ultracold Bose gas in a coordinate frame that adaptively scales with the system
size during evolution, enabling simulations of long evolution times during
expansion or similar large-scale manipulation. Our approach makes no
hydrodynamic approximations, is not restricted to a scaling ansatz, harmonic
potentials, or energy eigenstates, and can be generalized readily to
non-contact interactions via the appropriate stress tensor of the quantum
fluid. As applications, we simulate the expansion of the ideal gas, a
cigar-shaped condensate in the Thomas-Fermi regime, and a linear superposition
of counter propagating Gaussian wavepackets. We recover known scaling for the
ideal gas and Thomas-Fermi regimes, and identify a linear regime of
aspect-ratio preserving free expansion; analysis of the scaling dynamics
equations shows that an exact, aspect-ratio invariant, free expansion does not
exist for nonlinear evolution. Our treatment enables exploration of nonlinear
effects in matter-wave dynamics over large scale-changing evolution.
