Lighthill equation for quantum liquids

Abstract

A quantum version of the Lighthill equation that originated in the field of theoretical aeroacoustics is derived for the probability density of a superfluid starting from the time-dependent Gross-Pitaevskii equation. It involves a second-order time derivative and should be supplemented by two-time boundary conditions. Its physical implications are discussed in relation to the quantum equilibrium hypothesis and the general applicability of Born's rule.