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arXiv · 2608.14174

Ground States and Periodic--to--Localized Convergence in Two--Dimensional Saturable Discrete Nonlinear Schr\"odinger Equations

Vassilios M Rothos
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Abstract

We study a two--dimensional discrete nonlinear Schr\"odinger 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 $\ell^2(\mathbb Z^2)$. 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 $\ell^2(\mathbb Z^2)$ 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.

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Vassilios M Rothos. 2026-08-14. Ground States and Periodic--to--Localized Convergence in Two--Dimensional Saturable Discrete Nonlinear Schr\"odinger Equations. https://arxiv.org/abs/2608.14174

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