  We study the nonequilibrium dynamics of an electronic model of competition
between an unconventional charge density wave (a bond density wave) and
$d$-wave superconductivity. In a time-dependent Hartree-Fock+BCS approximation,
the dynamics reduces to the equations of motion of operators realizing the
generators of SU(4) at each pair of momenta, ( $\boldsymbol{k}$, -
$\boldsymbol{k}$ ), in the Brillouin zone. We also study the nonequilibrium
dynamics of a quantum generalization of a O(6) nonlinear $\sigma$ model of
competing orders in the underdoped cuprates [Hayward et al., Science
$\boldsymbol{343}$, 1336 (2014)]. We obtain results, in the large $N$ limit of
a O($N$) model, on the time dependence of correlation functions following a
pulse disturbance. We compare our numerical studies with recent picosecond
optical experiments. We find that, generically, the oscillatory responses in
our models share various qualitative features with the experiments.
