Frozen dynamics of a breather induced by an adiabatic invariant

TL;DR

This paper proves that the slow, frozen dynamics of breathers in a simplified DNLS model are due to an adiabatic invariant, which explains the stability and destabilization mechanisms of breathers.

Contribution

It provides a rigorous proof of the adiabatic invariant's role in frozen breather dynamics within a simplified DNLS framework, combining perturbative and phenomenological methods.

Findings

Identification of the adiabatic invariant as key to frozen dynamics

Sudden jumps destabilize breathers, resembling Lévy processes

Nonlinearity and deterministic forcing are essential for frozen behavior

Abstract

The Discrete Nonlinear Schr\"odinger (DNLS) equation is a Hamiltonian model displaying an extremely slow relaxation process when discrete breathers appear in the system. In [Iubini S, Chirondojan L, Oppo G L, Politi A and Politi P 2019 Physical Review Letters 122 084102], it was conjectured that the frozen dynamics of tall breathers is due to the existence of an adiabatic invariant (AI). Here, we prove the conjecture in the simplified context of a unidirectional DNLS equation, where the breather is "forced" by a background unaffected by the breather itself. We first clarify that the nonlinearity of the breather dynamics and the deterministic nature of the forcing term are both necessary ingredients for the existence of a frozen dynamics. We then derive perturbative expressions of the AI by implementing a canonical perturbation theory and via a more phenomenological approach based on the estimate of the energy flux. The resulting accurate identification of the AI allows revealing the presence and role of sudden jumps as the main breather destabilization mechanism, with an unexpected similarity with L\'evy processes.