Ground states and periodic-to-localized convergence in two-dimensional saturable discrete nonlinear Schrödinger equations

Abstract

We study a two-dimensional discrete nonlinear Schrödinger equation with saturable nonlinearity on the lattice Z2. 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 ℓ2(Z2). 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 ℓ2(Z2) 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.