  Motivated by the recent experimental progress in studying conventional and
unconventional superconductors in a pump-probe setup, we perform a
comprehensive theoretical analysis of the nonlinear response of a diffusive BCS
conventional superconductor to the action of an alternating electromagnetic
field using a generalized Usadel equation. We analyze the response up to the
second order of the perturbation in the amplitude of the vector potential
$\vec{A}$, the superconducting order parameter $\Delta$ and in the third order
for the current $\vec{j}$. On the basis of this approach, we derive general
expressions for the retarded (advanced) Green's functions, as well as the
Keldysh function for an arbitrary number of harmonics of the incident field.
Most importantly, we analyze the set of physical observables in a
non-equilibrium superconductor, such as frequency and the temperature
dependencies of the zero harmonic $\delta( \Delta)_0$ (Eliashberg effect), the
second harmonic $\delta( \Delta)_{2\Omega}$, as well as the third harmonic for
the electric current $j(3\Omega)$ under the action of a monochromatic
irradiation. For the same set of parameters, we also analyze the behavior of
the reflectivity and the down-conversion intensity of a thin superconducting
film, discussed recently in the context of parametric amplification of
superconductivity. We derive these quantities microscopically and show the
connection of the down-conversion intensity to the third harmonic generation
currents induced by the amplitude mode and the direct action of the electric
field on the charge carriers.
