Numerical spectral analysis of standing waves in quantum hydrodynamics with viscosity

TL;DR

This paper investigates the spectral stability of standing wave solutions in quantum hydrodynamics with viscosity, revealing potential instabilities through numerical analysis of eigenvalues.

Contribution

It introduces a numerical Evans function approach to analyze spectral stability in quantum hydrodynamics with viscosity, highlighting potential instabilities.

Findings

Essential spectrum is stable for the studied profiles.

Numerical evidence of a real unstable eigenvalue.

Indication of spectral instability in the systems.

Abstract

We study the spectrum of the linearization around standing wave profiles for two quantum hydrodynamics systems with linear and nonlinear viscosity. The essential spectrum for such profiles is stable; we investigate the point spectrum using an Evans function technique. For both systems we show numerically that there exists a real unstable eigenvalue, thus providing numerical evidence for spectral instability.