The Fine Structure of the Phonon in One Dimension from Quantum Hydrodynamics

We show that the resonant interactions between phonons in one dimension may be treated consistently within Quantum Hydrodynamics by the introduction of phonon dispersion. In this way the physics of a nonlinear Luttinger liquid may be described in terms of hydrodynamic (i.e. bosonized) variables without recourse to refermionization or the introduction of fictitious impurities. We focus on the calculation of the dynamic structure factor for a model with quadratic dispersion, which has the Benjamin--Ono equation of fluid dynamics as its equation of motion. We find singular behavior in the vicinity of upper and lower energetic thresholds corresponding to phonon and soliton branches of the classical theory, which may be benchmarked against known results for the Calogero--Sutherland model.