Metastability phenomena in two-dimensional rectangular lattices with nearest-neighbour interaction

TL;DR

This paper analytically investigates metastability in two-dimensional rectangular lattices with nearest-neighbour interactions, demonstrating that energy remains localized among low-frequency modes over significant timescales.

Contribution

It introduces a normal form approach linking lattice dynamics to integrable PDEs, revealing metastability phenomena in anisotropic initial conditions.

Findings

Energy distribution among low-frequency modes remains stable over time

Normal form reduces lattice dynamics to integrable PDEs

Metastability persists within the approximation's validity time

Abstract

We study analytically the dynamics of two-dimensional rectangular lattices with periodic boundary conditions. We consider anisotropic initial data supported on one low-frequency Fourier mode. We show that, in the continuous approximation, the resonant normal form of the system is given by integrable PDEs. We exploit the normal form result in order to prove the existence of metastability phenomena for the lattices. More precisely, we show that the energy spectrum of the normal modes attains a distribution in which the energy is shared among a packet of low-frequencies modes; such distribution remains unchanged up to the time-scale of validity of the continuous approximation.