  The Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) is designed to
detect and measure the redshifts of more than one million Ly$\alpha$ emitting
galaxies (LAEs) between $1.88 < z < 3.52$. In addition to its cosmological
measurements, these data enable studies of Ly$\alpha$ spectral profiles and the
underlying radiative transfer. Using the roughly half a million LAEs in the
HETDEX Data Release 3, we stack various subsets to obtain the typical
Ly$\alpha$ profile for the $z \sim 2-3$ epoch and to understand their physical
properties. We find clear absorption wings around Ly$\alpha$ emission, which
extend $\sim 2000$ km $\mathrm{s}^{-1}$ both redward and blueward of the
central line. Using far-UV spectra of nearby ($0.002 < z < 0.182$) LAEs in the
CLASSY treasury and optical/near-IR spectra of $2.8 < z < 6.7$ LAEs in the
MUSE-Wide survey, we observe absorption profiles in both redshift regimes.
Dividing the sample by volume density shows that the troughs increase in higher
density regions. This trend suggests that the depth of the absorption is
dependent on the local density of objects near the LAE, a geometry that is
similar to damped Lyman-$\alpha$ systems. Simple simulations of Ly$\alpha$
radiative transfer can produce similar troughs due to absorption of light from
background sources by HI gas surrounding the LAEs.
