Spectral analysis of dispersive shocks for quantum hydrodynamics with nonlinear viscosity

TL;DR

This paper analyzes the spectral stability of traveling wave solutions in a 1D quantum hydrodynamics model with nonlinear viscosity, combining analytical conditions with numerical Evans function computations.

Contribution

It provides a new spectral stability criterion for quantum hydrodynamics with nonlinear viscosity and numerically verifies the stability of the point spectrum.

Findings

Derived a sufficient condition for essential spectrum stability.

Estimated eigenvalues with non-negative real part.

Numerical Evans function analysis suggests point spectrum stability.

Abstract

In this paper we investigate spectral stability of traveling wave solutions to 1-$D$ quantum hydrodynamics system with nonlinear viscosity in the $(\rho,u)$, that is, density and velocity, variables. We derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of eigenvalues with non-negative real part. In addition, we present numerical computations of the Evans function in sufficiently large domain of the unstable half-plane and show numerically that its winding number is (approximately) zero, thus giving a numerical evidence of point spectrum stability.