  We present a first attempt to model the narrow-line (NL) region of active
galactic nuclei (AGN) in hydrodynamic simulations of galaxy mergers, using a
novel physical prescription. This model is used to determine the origin of
double-peaked NL (dNL) AGN in merging galaxies and their connection to
supermassive black hole (SMBH) pairs, motivated by recent observations of such
objects. We find that dNL AGN induced by the relative motion of SMBH pairs are
a generic but short-lived feature of gaseous major mergers. dNL AGN should
often be observed in late-stage mergers, during the kpc-scale phase of SMBH
inspiral or soon after the SMBH merger. However, even within the kpc-scale
phase, only a minority of dNL AGN are directly induced by SMBH motion; their
lifetimes are typically a few Myr. Most double peaks arise from gas kinematics
near the SMBH, although prior to the SMBH merger up to 80% of all dNL profiles
may be influenced by SMBH motion via altered peak ratios or velocity offsets.
The total lifetimes of dNL AGN depend strongly on viewing angle and on
properties of the merging galaxies. Also, in a typical merger, at least 10-40%
of the double peaks induced by SMBH motion have small projected separations,
0.1-1 kpc, such that dual peaks of stellar surface brightness are not easily
resolved. Diffuse tidal features can indicate late-stage galaxy mergers,
although they do not distinguish SMBH pairs from merged SMBHs. We show that dNL
profiles with peak velocity splittings > 500 km s^-1 or with measurable overall
velocity shifts are often associated with SMBH pairs. Our results support the
notion that selection of dNL AGN is a promising method for identifying dual
SMBH candidates, but demonstrate the critical importance of high-resolution,
multi-wavelength follow-up observations, and the use of multiple lines of
evidence, for confirming the dual nature of candidate SMBH pairs. (Abridged)
