  We introduce a model combining Kerr nonlinearity with a periodically changing
sign ("nonlinearity management") and a Bragg grating (BG). The main result,
obtained by means of systematic simulations, is presented in the form of a
soliton's stability diagram on the parameter plane of the model; the diagram
turns out to be a universal one, as it practically does not depend on the
soliton's power. Moreover, simulations of the nonlinear Schroedinger (NLS)
model subjected to the same "nonlinearity management" demonstrate that the same
diagram determines the stability of the NLS solitons, unless they are very
narrow. The stability region of very narrow NLS solitons is much smaller, and
soliton splitting is readily observed in that case. The universal diagram shows
that a minimum non-zero average value of the Kerr coefficient is necessary for
the existence of stable solitons. Interactions between identical solitons with
an initial phase difference between them are simulated too in the BG model,
resulting in generation of stable moving solitons. A strong spontaneous
symmetry breaking is observed in the case when in-phase solitons pass through
each other due to attraction between them.
