Frozen dynamics of a breather induced by an adiabatic invariant

Abstract

The discrete nonlinear Schrödinger (DNLS) equation is a Hamiltonian model displaying an extremely slow relaxation process when discrete breathers appear in the system. In (Iubini et al 2019 Phys. Rev. Lett. 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 Lévy processes.

Intensity pseudo-localized phase in the glassy random laser

“…As an instance, this behavior has been found and very precisely described in the framework of large deviation calculations and ensemble inequivalence in the case of mass-transport models [39,40] or for bosonic condensates in optical lattices described by the discrete non-linear Schrödinger equation (DNLSE) [41]. Concerning the DNLSE, it is known since almost two decades that his high energy phase is characterized by the appearance of localized breather-like solutions, which typically arise in certain regimes in models of non-linear waves [33,[42][43][44][45][46][47].…”