# Relaxation and coarsening of weakly-interacting breathers in a simplified DNLS chain

## TL;DR

This paper investigates the slow relaxation and freezing of multi-breather configurations in the DNLS equation by introducing stochastic models that mimic the weak coupling and coarsening dynamics of breathers, revealing regimes of slowing down and freezing.

## Contribution

It introduces simplified stochastic models to study breather interactions and demonstrates how coupling decay rates influence coarsening and freezing phenomena in DNLS-like systems.

## Key findings

- Coarsening slows down with faster decay rates of coupling.
- Exponential decay leads to a freezing effect similar to DNLS.
- Weak coupling causes multi-breather configurations to become effectively frozen.

## Abstract

The Discrete NonLinear Schr\"odinger (DNLS) equation displays a parameter region characterized by the presence of localized excitations (breathers). While their formation is well understood and it is expected that the asymptotic configuration comprises a single breather on top of a background, it is not clear why the dynamics of a multi-breather configuration is essentially frozen. In order to investigate this question, we introduce simple stochastic models, characterized by suitable conservation laws. We focus on the role of the coupling strength between localized excitations and background. In the DNLS model, higher breathers interact more weakly, as a result of their faster rotation. In our stochastic models, the strength of the coupling is controlled directly by an amplitude-dependent parameter. In the case of a power-law decrease, the associated coarsening process undergoes a slowing down if the decay rate is larger than a critical value. In the case of an exponential decrease, a freezing effect is observed that is reminiscent of the scenario observed in the DNLS. This last regime arises spontaneously when direct energy diffusion between breathers and background is blocked below a certain threshold.