Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

TL;DR

This paper proves the spectral stability of small-amplitude dispersive shock profiles in a quantum hydrodynamics model with viscosity, confirming previous numerical findings through spectral energy estimates.

Contribution

It provides the first rigorous proof of spectral stability for small-amplitude viscous-dispersive shocks in quantum hydrodynamics, using spectral energy estimates.

Findings

Spectral stability of small-amplitude shocks is established.

The proof confirms previous numerical observations.

Monotonicity of profiles is key to the analysis.

Abstract

A compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The dispersive term is due to quantum effects described through the Bohm potential and the viscosity term is of linear type. It is shown that small-amplitude viscous-dispersive shock profiles for the system under consideration are spectrally stable, proving in this fashion a previous numerical observation by Lattanzio et al. (Phys. D 402, 2020, p. 132222; Appl. Math. Comput. 385, 2020, p. 125450). The proof is based on spectral energy estimates which profit from the monotonicty of the profiles in the small-amplitude regime.