  We study the time evolution of the mass gap of the O(N) non-linear sigma
model in 2+1 dimensions due to a time-dependent coupling in the large-$N$
limit. Using the Schwinger-Keldysh approach, we derive a set of equations at
large $N$ which determine the time dependent gap in terms of the coupling.
These equations lead to a criterion for the breakdown of adiabaticity for slow
variation of the coupling leading to a Kibble-Zurek scaling law. We describe a
self-consistent numerical procedure to solve these large-$N$ equations and
provide explicit numerical solutions for a coupling which starts deep in the
gapped phase at early times and approaches the zero temperature equilibrium
critical point $g_c$ in a linear fashion. We demonstrate that for such a
protocol there is a value of the coupling $g= g_c^{\rm dyn}> g_c$ where the gap
function vanishes, possibly indicating a dynamical instability. We study the
dependence of $g_c^{\rm dyn}$ on both the rate of change of the coupling and
the initial temperature. We also verify, by studying the evolution of the mass
gap subsequent to a sudden change in $g$, that the model does not display
thermalization within a finite time interval $t_0$ and discuss the implications
of this observation for its conjectured gravitational dual as a higher spin
theory in $AdS_4$.
