  We address existence of moving gap solitons (traveling localized solutions)
in the Gross-Pitaevskii equation with a small periodic potential. Moving gap
solitons are approximated by the explicit localized solutions of the
coupled-mode system. We show however that exponentially decaying traveling
solutions of the Gross-Pitaevskii equation do not generally exist in the
presence of a periodic potential due to bounded oscillatory tails ahead and
behind the moving solitary waves. The oscillatory tails are not accounted in
the coupled-mode formalism and are estimated by using techniques of spatial
dynamics and local center-stable manifold reductions. Existence of bounded
traveling solutions of the Gross--Pitaevskii equation with a single bump
surrounded by oscillatory tails on a finite large interval of the spatial scale
is proven by using these technique. We also show generality of oscillatory
tails in other nonlinear equations with a periodic potential.
