  We investigate the dynamics of an effectively one-dimensional Bose-Einstein
condensate (BEC) with scattering length $a$ subjected to a spatially periodic
modulation, $a=a(x)=a(x+L)$. This "collisionally inhomogeneous" BEC is
described by a Gross-Pitaevskii (GP) equation whose nonlinearity coefficient is
a periodic function of $x$. We transform this equation into a GP equation with
constant coefficient $a$ and an additional effective potential and study a
class of extended wave solutions of the transformed equation. For weak
underlying inhomogeneity, the effective potential takes a form resembling a
superlattice, and the amplitude dynamics of the solutions of the
constant-coefficient GP equation obey a nonlinear generalization of the Ince
equation. In the small-amplitude limit, we use averaging to construct
analytical solutions for modulated amplitude waves (MAWs), whose stability we
subsequently examine using both numerical simulations of the original GP
equation and fixed-point computations with the MAWs as numerically exact
solutions. We show that "on-site" solutions, whose maxima correspond to maxima
of $a(x)$, are significantly more stable than their "off-site" counterparts.
