Thermal solitons as revealed by static structure factor

We study, within a framework of the classical fields approximation, the static structure factor of a weakly interacting Bose gas at thermal equilibrium. As in a recent experiment (R. Schley et al., Phys. Rev. Lett. 111, 055301 (2013)), we find that the thermal distribution of phonons in a three-dimensional Bose gas follows the Planck distribution. On the other hand we find a disagreement between the Planck and phonon (calculated just like for the bulk gas) distributions in the case of elongated quasi-one-dimensional systems. We attribute this discrepancy to the existence of spontaneous dark solitons (i.e., thermal solitons as reported in T. Karpiuk et al., Phys. Rev. Lett. 109, 205302 (2012)) in an elongated Bose gas at thermal equilibrium.