Superfluidity versus Disorder in the Discrete Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates how superfluidity and disorder affect wave propagation in the discrete nonlinear Schrödinger equation, revealing a superfluid regime where waves coherently traverse defects, with implications for BECs and optical fibers.

Contribution

It introduces a superfluidity criterion for the DNLS in disordered systems, analogous to Landau's criterion, and analyzes the transition between different dynamical regimes.

Findings

Superfluid regime allows coherent wave transmission through defects.

Critical nonlinearity threshold for superfluidity is identified.

Different dynamical regimes include reflection, refocusing, and solitonic behavior.

Abstract

We study the discrete nonlinear Schr\"odinger equation (DNLS) in an annular geometry with on-site defects. The dynamics of a traveling plane-wave maps onto an effective ''non-rigid pendulum'' Hamiltonian. The different regimes include the complete reflection and refocusing of the initial wave, solitonic structures, and a superfluid state. In the superfluid regime, which occurs above a critical value of nonlinearity, a plane-wave travels coherently through the randomly distributed defects. This superfluidity criterion for the DNLS is analogous to (yet very different from) the Landau superfluidity criteria in translationally invariant systems. Experimental implications for the physics of Bose-Einstein condensate gases trapped in optical potentials and of arrays of optical fibers are discussed.