  Second-harmonic generation (SHG) is a well-known nonlinear spectroscopy
method to probe electronic structure, specifically, in transition metal
dichalcogenide (TMDC) monolayers. This work investigates the nonlinear dynamics
of a strongly excited TMDC monolayer by solving the time evolution equations
for the density matrix. It is shown that the presence of excitons qualitatively
changes the nonlinear dynamics leading, in particular, to a huge enhancement of
the nonlinear signal as a function of the dielectric environment. It is also
shown that the SHG polarization angular diagram and its dependence on the
driving strength are very sensitive to the type of exciton state. This
sensitivity suggests that SHG spectroscopy is a convenient tool for analyzing
the fine structure of excitonic states.
