Numerical investigations of dispersive shocks and spectral analysis for linearized quantum hydrodynamics

Abstract

Abstract The aim of this paper is to study solutions of one dimensional compressible Euler system with dissipation–dispersion terms, where the dispersive term is originated by the quantum effects described through the Bohm potential, as customary in quantum hydrodynamic models. We shall investigate numerically the sensitivity of the profiles with respect to the viscosity parameter, in particular in terms of their monotonicity properties. In addition, we shall also pinpoint numerically how the profile becomes more oscillatory as the end states approach the vacuum. The analysis of spectral properties of the linearized system around constant states is also provided, as well as the (numerical) localization of the point spectrum of the linearization along a profile. The latter investigation is carried out through the Evans function method.