  Annular substructures in protoplanetary discs, ubiquitous in sub-mm
observations, can be caused by gravitational coupling between a disc and its
embedded planets. Planetary density waves inject angular momentum into the disc
leading to gap opening only after travelling some distance and steepening into
shocks (in the absence of linear damping); no angular momentum is deposited in
the planetary coorbital region, where the wave has not shocked yet. Despite
that, simulations show mass evacuation from the coorbital region even in
inviscid discs, leading to smooth, double-trough gap profiles. Here we consider
the early, time-dependent stages of planetary gap opening in inviscid discs. We
find that an often-overlooked contribution to the angular momentum balance
caused by the time-variability of the specific angular momentum of the disc
fluid (caused, in turn, by the time-variability of the radial pressure support)
plays a key role in gap opening. Focusing on the regime of shallow gaps with
depths of $\lesssim 20\%$, we demonstrate analytically that early gap opening
is a self-similar process, with the amplitude of the planet-driven perturbation
growing linearly in time and the radial gap profile that can be computed
semi-analytically. We show that mass indeed gets evacuated from the coorbital
region even in inviscid discs. This evolution pattern holds even in viscous
discs over a limited period of time. These results are found to be in excellent
agreement with 2D numerical simulations. Our simple gap evolution solutions can
be used in studies of dust dynamics near planets and for interpreting
protoplanetary disc observations.
