  Recently, several groups have reported observations of collective modes of
the charge order present in underdoped cuprates. Motivated by these
experiments, we study theoretically the oscillations of the order parameters,
both in the case of pure charge order, and for charge order coexisting with
superconductivity. Using a hot-spot approximation we find in the coexistence
regime two Higgs modes arising from hybridization of the amplitude oscillations
of the different order parameters. One of them has a minimum frequency that is
within the single particle energy gap and which is a non-monotonic function of
temperature. The other -- high-frequency -- mode is smoothly connected to the
Higgs mode in the single-order-parameter region, but quickly becomes overdamped
in the case of coexistence. We explore an unusual low-energy damping channel
for the collective modes, which relies on the band reconstruction caused by the
coexistence of the two orders. For completeness, we also consider the damping
of the collective modes originating from the nodal quasiparticles. At the end
we discuss some experimental consequences of our results.
