Instabilities in the Bogoliubov Spectrum of a condensate in a 1-D periodic potential

TL;DR

This paper analyzes the stability of standing wave solutions in a 1-D Gross-Pitaevsky equation with a periodic potential, providing a rigorous criterion for exponential instability based on spectral analysis.

Contribution

It introduces a simple, rigorous spectral criterion for instability of standing waves in a 1-D periodic Gross-Pitaevsky system, linking it to the spectrum of a self-adjoint operator.

Findings

Derived a criterion for instability based on spectral properties.

Confirmed the criterion simplifies for small amplitude solutions.

Connected the instability criterion to effective mass concepts.

Abstract

We study the stability of standing wave solutions to a one-dimensional Gross-Pitaevsky equation with a periodic potential. We use some simple complex analysis and the Hamiltonian structure of the problem to give a simple rigorous criterion which guarantees the existence of non-real spectrum, which corresponds to exponential instability of the standing wave solution. This criterion can be stated simply in terms of the spectrum of one of these self-adjoint operators. When the standing wave has small amplitude this criterion simplifies further, and agrees with arguments based on the effective mass in the periodic potential.