  We construct a class of exact commensurate and incommensurate standing wave
(SW) solutions in a piecewise smooth analogue of the discrete non-linear
Schr\"{o}dinger (DNLS) model and present their linear stability analysis. In
the case of the commensurate SW solutions the analysis reduces to the
eigenvalue problem of a transfer matrix depending parametrically on the
eigenfrequency. The spectrum of eigenfrequencies and the corresponding
eigenmodes can thereby be determined exactly. The spatial periodicity of a
commensurate SW implies that the eigenmodes are of the Bloch form,
characterised by an even number of Floquet multipliers. The spectrum is made up
of bands that, in general, include a number of transition points corresponding
to changes in the disposition of the Floquet multipliers. The latter
characterise the different band segments. An alternative characterisation of
the segments is in terms of the Krein signatures associated with the
eigenfrequencies. When one or more parameters characterising the SW solution is
made to vary, one occasionally encounters collisions between the band-edges or
the intra-band transition points and, depending on the the Krein signatures of
the colliding bands or segments, the spectrum may stretch out in the complex
plane, leading to the onset of instability. We elucidate the correlation
between the disposition of Floquet multipliers and the Krein signatures,
presenting two specific examples where the SW possesses a definite window of
stability, as distinct from the SW's obtained close to the anticontinuous and
linear limits of the DNLS model.
