  We present eight types of spatial optical solitons which are possible in a
model of a planar waveguide that includes a dual-channel trapping structure and
competing (cubic-quintic) nonlinearity. Among the families of trapped beams are
symmetric and antisymmetric solitons of "broad" and "narrow" types, composite
states, built as combinations of broad and narrow beams with identical or
opposite signs ("unipolar" and "bipolar" states, respectively), and
"single-sided" broad and narrow beams trapped, essentially, in a single
channel. The stability of the families is investigated via eigenvalues of small
perturbations, and is verified in direct simulations. Three species - narrow
symmetric, broad antisymmetric, and unipolar composite states - are unstable to
perturbations with real eigenvalues, while the other five families are stable.
The unstable states do not decay, but, instead, spontaneously transform
themselves into persistent breathers, which, in some cases, demonstrate
dynamical symmetry breaking and chaotic internal oscillations. A noteworthy
feature is a stability exchange between the broad and narrow antisymmetric
states: in the limit when the two channels merge into one, the former species
becomes stable, while the latter one loses its stability. Different branches of
the stationary states are linked by four bifurcations, which take different
forms in the model with the strong and weak inter-channel coupling.
