Non-linear acoustics in Galilean and relativistic barotropic fluids

Abstract

Non-linear waves described by the defocusing non-linear Schroedinger (NLS) equation admit a hydrodynamical representation in terms of Galilean potential flows and, using this correspondence, an autonomous equation for potential flow's non-linear acoustic has been recently derived by Nore et al. However, this equation does not contain simple solutions of the original one such as (dark) solitons. The purpose of the present article is to characterize the reasons behind this failure and to present an original method to build separate equations describing all different types of acoustic solutions (but one). For reasons of generality, we work in a framework adapted to special relativistic hydrodynamics. All the results we derive have Galilean counterparts which are also discussed. In particular, we argue that there exist an infinity of different acoustic sectors for relativistic barotropic fluids, and we prove this result for fluids with a particularly simple equation of state. Solitons are naturally captured by our approach and a few explicit examples are worked out. Conserved quantities for the acoustic regime are also derived.