Finite-temperature hydrodynamics for one-dimensional Bose gases:   Breathing mode oscillations as a case study
  We develop a finite-temperature hydrodynamic approach for a harmonically trapped one-dimensional quasicondensate and apply it to describe the phenomenon of frequency doubling in the breathing-mode oscillations of its momentum distribution. The doubling here refers to the oscillation frequency relative to the oscillations of the real-space density distribution, invoked by a sudden confinement quench. We find that the frequency doubling is governed by the quench strength and the initial temperature, rather than by the crossover from the ideal Bose gas to the quasicondensate regime. The hydrodynamic predictions are supported by the results of numerical simulations based on a finite-temperature c-field approach, and extend the utility of the hydrodynamic theory for low-dimensional quantum gases to the description of finite-temperature systems and their dynamics in momentum space. 
I. Bouchoule, S. S. Szigeti, M. J. Davis, and K. V. Kheruntsyan
cond-mat.quant-gas quant-ph