  In this paper we study existence and stability of shock profiles for a 1-D
compressible Euler system in the context of Quantum Hydrodynamic models. The
dispersive term is originated by the quantum effects described through the Bohm
potential; moreover we introduce a (linear) viscosity to analyze its interplay
with the former while proving existence, monotonicity and stability of
travelling waves connecting a Lax shock for the underlying Euler system. The
existence of monotone profiles is proved for sufficiently small shocks; while
the case of large shocks leads to the (global) existence for an oscillatory
profile, where dispersion plays a significant role. The spectral analysis of
the linearized problem about a profile is also provided. In particular, we
derive a sufficient condition for the stability of the essential spectrum and
we estimate the maximum modulus of the eigenvalues in the unstable plane, using
a careful analysis of the Evans function.
