  We formulate a dynamical model to describe a photo-induced charge density
wave (CDW) quench transition and apply it to recent multi-probe experiments on
LaTe$_3$ [A. Zong et al., Nat. Phys. 15, 27 (2019)]. Our approach relies on
coupled time-dependent Ginzburg-Landau equations tracking two order parameters
that represent the modulations of the electronic density and the ionic
positions. We aim at describing the amplitude of the order parameters under the
assumption that they are homogeneous in space. This description is supplemented
by a three-temperature model, which treats separately the electronic
temperature, temperature of the lattice phonons with stronger couplings to the
electronic subsystem, and temperature of all other phonons. The broad scope of
available data for LaTe$_3$ and similar materials as well as the synergy
between different time-resolved spectroscopies allow us to extract model
parameters. The resulting calculations are in good agreement with ultra-fast
electron diffraction experiments, reproducing qualitative and quantitative
features of the CDW amplitude evolution during the initial few picoseconds
after photoexcitation.
