  Motivated by recent proposals of ``collisionally inhomogeneous''
Bose-Einstein condensates (BECs), which have a spatially modulated scattering
length, we study the existence and stability properties of bright and dark
matter-wave solitons of a BEC characterized by a periodic, piecewise-constant
scattering length. We use a ``stitching'' approach to analytically approximate
the pertinent solutions of the underlying nonlinear Schr\"odinger equation by
matching the wavefunction and its derivatives at the interfaces of the
nonlinearity coefficient. To accurately quantify the stability of bright and
dark solitons, we adapt general tools from the theory of perturbed Hamiltonian
systems. We show that solitons can only exist at the centers of the constant
regions of the piecewise-constant nonlinearity. We find both stable and
unstable configurations for bright solitons and show that all dark solitons are
unstable, with different instability mechanisms that depend on the soliton
location. We corroborate our analytical results with numerical computations.
