  We study the nonlinear dynamical evolution of spinodal decomposition in a
first-order superfluid phase transition using a simple holographic model in the
probe limit. We first confirm the linear stability analysis based on
quasinormal modes and verify the existence of a critical length scale related
to a gradient instability -- negative speed of sound squared -- of the
superfluid sound mode, which is a consequence of a negative thermodynamic
charge susceptibility. We present a comparison between our case and the
standard Cahn-Hilliard equation for spinodal instability, in which a critical
length scale can be also derived based on a diffusive instability. We then
perform several numerical tests which include the nonlinear time evolution
directly from an unstable state and fast quenches from a stable to an unstable
state in the spinodal region. Our numerical results provide a real time
description of spinodal decomposition and phase separation in one and two
spatial dimensions. We reveal the existence of four different stages in the
dynamical evolution, and characterize their main properties. Finally, we
investigate the strength of dynamical heterogeneity using the spatial variance
of the local chemical potential and we correlate the latter to other features
of the dynamical evolution.
