Boris Malomed Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Israel Finite-band solitons in the Kronig-Penney model with the cubic-quintic nonlinearity Tuesday 7th, September 14:05-14:55pm, Eastern Avenue 310. We introduce a model combining a periodic array of rectangular potential wells [the Kronig-Penney (KP) potential] and the cubic- quintic (CQ) nonlinearity. A plethora of soliton states is found in the system: fundamental single-humped solitons, symmetric and antisymmetric double-humped ones, three-peak solitons with and without the phase shift between the peaks, etc. If the potential is weak, the solitons belong to the semi-infinite gap beneath the band structure of the linear KP model, while finite gaps between the Bloch bands remain empty. However, in contrast with the situation known in the model combining a periodic potential and the self-focusing Kerr nonlinearity, the solitons fill only a finite zone near the top of the semi-infinite gap, which is a manifestation of the saturable character of the CQ nonlinearity. If the potential is stronger, fundamental and double (both symmetric and antisymmetric) solitons with a flat-top shape are also found in the finite gaps. Computation of stability eigenvalues for small perturbations and direct simulations show that all the solitons are stable. The soliton characteristics, in the form of the integral power Q (or width w) vs. the propagation constant k, reveal strong bistability, with two and, sometimes, four different solutions found for given k. Disobeying the Vakhitov-Kolokolov criterion, the solution branches with both dQ/dk > 0 and dQ/dk < 0 are stable. Another distinctive feature of the model is its beam-splitting property: while the amplitude of the solitons is limited, increase of the integral power gives rise to additional peaks in the soliton's shape, each corresponding to a sub-pulse trapped in an a local channel of the KP structure. It is plausible that these features are shared by other models combining a saturable nonlinearity and a periodic substrate.