Existence of standing and traveling waves in quantum hydrodynamics with viscosity

Abstract

In this paper, we consider two quantum hydrodynamics systems with linear and nonlinear viscosity. For the former, we prove existence of standing waves; moreover, global existence of traveling waves is proved 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 by Lattanzio, Marcati, and Zhelyazov [Phys. D 402 (2020), 132222], where it was required that the viscosity is sufficiently strong. In addition, we discuss existence of standing waves for the latter system.