Thermalization in a nonlinear variant of the discrete nonlinear Schr\"odinger Equation

Abstract

We study the thermalization properties of a fully nonlinear lattice model originally derived from the two-dimensional cubic defocusing nonlinear Schr{\"o}dinger 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 $\sim T^{-1/2}$ decay toward the faster $\sim T^{-1}$ decay characteristic of ergodic thermalization, where $T$ denotes the averaging time. In the high-energy regime, weak coupling $D\leq 1$ 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.