Ground States and Periodic--to--Localized Convergence in Two--Dimensional Saturable Discrete Nonlinear Schrödinger Equations

We study a two–dimensional discrete nonlinear Schrödinger equation with saturable nonlinearity on the lattice ${\mathbb{Z}}^2$. Using a variational approach based on the Nehari manifold, we establish the existence of nontrivial periodic ground states on finite lattices and establish the existence of exponentially localized ground states in ${\mathrm{\ell }}^2({\mathbb{Z}}^2)$. A principal result is the rigorous passage from periodic to localized states: we show that, up to lattice translations, periodic ground states converge strongly in ${\mathrm{\ell }}^2({\mathbb{Z}}^2)$ to a localized ground state as the lattice periods tend to infinity. The analysis combines variational methods, spectral properties of the discrete Laplacian, and concentration–compactness techniques adapted to the two–dimensional discrete setting. We further derive qualitative properties of the resulting solutions, including positivity and exponential localization, and establish a conditional orbital stability result within the Grillakis–Shatah–Strauss framework. Numerical computations illustrate the theoretical results and confirm the predicted convergence and localization behavior.