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A compressible viscous-dispersive Euler system in one space dimension in thecontext of quantum hydrodynamics is considered. The dispersive term is due toquantum effects described through the Bohm potential and the viscosity term isof linear type. It is shown that small-amplitude viscous-dispersive shockprofiles for the system under consideration are spectrally stable, proving inthis 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 basedon spectral energy estimates which profit from the monotonicty of the profilesin the small-amplitude regime.

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