  We introduce a model of three parallel-coupled nonlinear waveguiding cores
equipped with Bragg gratings (BGs), which form an equilateral triangle. The
objective of the work is to investigate solitons and their stability in this
system. New results are also obtained for the earlier investigated dual-core
system. Families of symmetric and antisymmetric solutions are found
analytically, extending beyond the spectral gap in both the dual- and tri-core
systems. Moreover, these families persist in the case (strong coupling between
the cores) when there is no gap in the system's linear spectrum. Three
different types of asymmetric solitons are found in the tri-core system. They
exist only inside the spectral gap, but asymmetric solitons with nonvanishing
tails are found outside the gap as well. The symmetric solitons are stable up
to points at which two types of asymmetric solitons bifurcate from them. Beyond
the bifurcation, one type of the asymmetric solitons is stable, and the other
is not. Then, they swap their stability. In both the dual- and tri-core
systems, the stability region of the symmetric solitons extends far beyond the
gap, persisting in the case when the system has no gap at all. The whole
stability region of antisymmetric solitons is located outside the gap. Thus,
solitons in multi-core BGs can be observed experimentally in a much broader
frequency band than in the single-core one, and in a wider parameter range than
it could be expected.
