Existence of standing and traveling waves in quantum hydrodynamics with viscosity

TL;DR

This paper proves the existence of standing and traveling wave solutions in quantum hydrodynamics systems with viscosity, improving previous results by removing restrictions on viscosity and dispersion parameters.

Contribution

It establishes the existence of both standing and traveling waves in quantum hydrodynamics with viscosity, including a global existence result for traveling waves without viscosity restrictions.

Findings

Existence of standing waves in quantum hydrodynamics with viscosity.

Global existence of traveling waves without viscosity restrictions.

Improvement over previous results requiring strong viscosity.

Abstract

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.