Traveling waves for quantum hydrodynamics with nonlinear viscosity

Abstract: In this paper we study existence of traveling waves for 1-D compressible Euler system with dispersion (which models quantum effects through the Bohm potential) and nonlinear viscosity in the context of quantum hydrodynamic models for superfluidity. The existence of profiles is proved for appropriate (super- or sub- sonic) end states defining Lax shocks for the underlying Euler system formulated in terms of density and velocity without restrictions for the viscosity and dispersion parameters. On the other hand, the interplay of the dispersion and the viscosity plays a crucial role in proving the existence of oscillatory profiles, showing in this way how the dispersion plays a significant role in certain regimes. Numerical experiments are also provided to analyze the sensitivity of such profiles with respect to the viscosity/dispersion terms and with respect to the nearness to vacuum.