Spectral stability of weak dispersive shock profiles for quantum hydrodynamics with nonlinear viscosity

TL;DR

This paper proves that small-amplitude weak dispersive shock profiles in quantum hydrodynamics with nonlinear viscosity are spectrally stable, combining analytical energy estimates with numerical spectrum validation.

Contribution

It provides the first rigorous proof of spectral stability for these shock profiles under small amplitude conditions.

Findings

Spectral stability holds for sufficiently small shock amplitudes.

Energy estimates confirm stability analytically.

Numerical results support the analytical findings.

Abstract

This paper studies the stability of weak dispersive shock profiles for a quantum hydrodynamics system in one space dimension with nonlinear viscosity and dispersive (quantum) effects due to a Bohm potential. It is shown that, if the shock amplitude is sufficiently small, then the profiles are spectrally stable. This analytical result is consistent with numerical estimations of the location of the spectrum (Lattanzio, Zhelyazov, Math. Models Methods Appl. Sci. 31, 2021). The proof is based on energy estimates at the spectral level, on the choice of an appropriate weighted energy function for the perturbations involving both the dispersive potential and the nonlinear viscosity, and on the montonicity of the dispersive profiles in the small-amplitude regime.