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