  We prove existence of standing waves for two quantum hydrodynamics systems
with linear and nonlinear viscosity. Moreover, global existence of traveling
waves is proved for the former without restrictions on the viscosity and
dispersion parameters, thanks to a suitable Lyapunov function. This is an
improvement with respect to the global existence result in \cite{LMZ2020},
where it was required that the viscosity is sufficiently strong.

Analysis of PDEs