  We present new results on the Ly-alpha emission-line kinematics of 18 z~2-3
star-forming galaxies with multiple-peaked Ly-alpha profiles. With our large
spectroscopic database of UV-selected star-forming galaxies at these redshifts,
we have determined that ~30% of such objects with detectable Ly-alpha emission
display multiple-peaked emission profiles. These profiles provide additional
constraints on the escape of Ly-alpha photons due to the rich velocity
structure in the emergent line. Despite recent advances in modeling the escape
of Ly-alpha from star-forming galaxies at high redshifts, comparisons between
models and data are often missing crucial observational information. Using Keck
II NIRSPEC spectra of H-alpha (z~2) and [OIII] 5007 (z~3), we have measured
accurate systemic redshifts, rest-frame optical nebular velocity dispersions
and emission-line fluxes for the objects in the sample. Accurate systemic
redshifts allow us to translate the multiple-peaked Ly-alpha profiles into
velocity space, revealing that the majority (11/18) display double-peaked
emission straddling the velocity-field zeropoint with stronger red-side
emission. We compare our data with the predictions of simple models for
outflowing and infalling gas distributions around high-redshift galaxies. While
popular "shell" models provide a qualitative match with many of the
observations of Ly-alpha emission, we find that in detail there are important
discrepancies between the models and data, as well as problems with applying
the framework of an expanding thin shell of gas to explain high-redshift galaxy
spectra. Our data highlight these inconsistencies, as well as illuminating
critical elements for success in future models of outflow and infall in
high-redshift galaxies. [Abridged]
