Solitons and their clusters sustained in saturable cubic-quintic nonlinear lattices.

We systematically investigate one- and two-dimensional (1D and 2D) solitons in periodically modulated media with saturable cubic and quintic nonlinearities, governed by a modified nonlinear Schrödinger equation. We discuss two distinct nonlinear regimes: purely focusing cubic-quintic media and inverted cubic-quintic media-featuring defocusing cubic and focusing quintic terms. Using the modified squared-operator method and Fourier collocation method, we analyze the existence domains, waveforms, powers and propagation constant relationships, and linear stability of fundamental solitons and their clusters. Our findings demonstrate that the combined action of saturable nonlinearities and nonlinear lattices effectively suppresses critical collapse and stabilizes diverse soliton structures, without relying on external linear periodic potentials or Bloch energy band theory. In both 1D and 2D systems, single solitons and their clusters tend to stabilize at larger propagation constants, with clusters exhibiting superior stability. Furthermore, the saturation coefficients S1 and S2 regulate the existence ranges, imposing strong saturation constraints on the propagation constants. In the competing (inverted) cubic-quintic nonlinearities, stable solutions exist only when S2 > S1. These results enrich the understanding of stable nonlinear localized modes in composite media with saturable nonlinearities and offer a theoretical framework for controlling optical localized modes in complex media, with potential applications in all-optical communication and information processing.