Stationary states of a nonlinear Schr\"odinger lattice with a harmonic trap

TL;DR

This paper investigates the stationary states of a nonlinear Schrödinger lattice with a harmonic trap, revealing how linear states evolve into nonlinear dark solitons, analyzing their stability, and exploring bifurcation structures.

Contribution

It provides a rigorous analytical and numerical study of nonlinear states in a trapped discrete Schrödinger lattice, including stability analysis and bifurcation structure.

Findings

Ground state is stable; excited states have stability/instability bands.

Discreteness can destabilize dark solitons, which are stable in the continuum limit.

Rich bifurcation structure is revealed from the anti-continuum limit.

Abstract

We study a discrete nonlinear Schr\"odinger lattice with a parabolic trapping potential. The model, describing, e.g., an array of repulsive Bose-Einstein condensate droplets confined in the wells of an optical lattice, is analytically and numerically investigated. Starting from the linear limit of the problem, we use global bifurcation theory to rigorously prove that - in the discrete regime - all linear states lead to nonlinear generalizations thereof, which assume the form of a chain of discrete dark solitons (as the density increases). The stability of the ensuing nonlinear states is studied and it is found that the ground state is stable, while the excited states feature a chain of stability/instability bands. We illustrate the mechanisms under which discreteness destabilizes the dark-soliton configurations, which become stable only inside the continuum regime. Continuation from the anti-continuum limit is also considered, and a rich bifurcation structure is revealed.