Thermalization in a nonlinear variant of the discrete nonlinear Schrödinger equation

We study the thermalization properties of a fully nonlinear lattice model originally derived from the two-dimensional cubic defocusing nonlinear Schrödinger equation (NLS) using analytical and numerical methods. The model conserves both energy and norm, whose densities define a microcanonical energy-norm parameter space, while the nonlinear nearest-neighbor coupling is controlled by a parameter D. Within this space, our analysis identifies broad parameter regimes in which the dynamics is ergodic not only within but also outside the standard Gibbs region, indicating the need for a modified statistical description. At higher energies, the system instead exhibits long-lived compacton-mediated localization and signatures of weak nonergodicity, as evidenced by finite-time variances, excursion-time statistics, and probability distributions of local amplitudes. We show that stronger coupling D enhances fluctuations and accelerates the crossover of the finite-time variance of the local norm density from an initial ∼T^{-1/2} decay toward the faster ∼T^{-1} decay characteristic of ergodic thermalization, where T denotes the averaging time. In the high-energy regime, weak coupling D≤1 favors persistent single-site compacton localization, whereas stronger coupling D>1 yields long-lived two-site localization. Our results provide insights into the interplay between thermalization, localization, and non-Gibbs statistical behavior in genuinely nonlinear systems.