  We calculate the ferroelectric polarization dynamics induced by a femtosecond
midinfrared pulse as measured in the recent experiment by R. Mankowsky et al.,
Phys. Rev. Lett. 118, 197601 (2017). It is due to the nonlinear coupling of the
excited infrared-active phonon with the ferroelectric mode or to the excitation
of the ferroelectric mode itself depending on the pulse frequency. To begin
with, we write the LiNbO$_3$ crystal symmetry invariant thermodynamic potential
including electric field and nonlinear phonon coupling terms. We solve the
equations of motion determined by this potential for phonon coordinates
numerically in classical approximation. We explain the transient polarization
reversal observed in the experiment by action of the depolarizing electric
field which is due to bound charges at the polarization domain boundaries and
give a reasonable estimate for its value. We argue that the polarization could
be ultimately reversed when this field is screened.
