Subsonic-or-sonic criterion for spectral stability of QHD wave profiles

A QHD wave profile consists of a density and velocity pair $(R,U)$ for a standing or traveling wave of the systems with linear or nonlinear viscosity. Let $c_s(R)$ denote the speed of sound at density $R$; the profile is subsonic or sonic when

$$|U(x)|\leq c_s(R(x)),\qquad x\in\mathbb{R}.$$

Subsonic-or-sonic stability conjecture. A standing or traveling wave profile for the QHD systems with linear or nonlinear viscosity is spectrally stable if and only if its velocity is subsonic or sonic along the profile, equivalently,

$$|U(x)|\leq c_s(R(x)),\qquad x\in\mathbb{R}.$$

The numerical results in the paper show spectral instability for the considered standing waves, while prior numerical results establish spectral stability for certain non-monotone traveling waves whose velocities are subsonic along the profiles. The conjecture is also consistent with the necessary and sufficient condition for stability of the essential spectrum, since the pointwise inequality implies subsonic-or-sonic end states $|U^{\pm}|\leq c_s(R^{\pm})$.