Excitations and thermodynamics in a one-dimensional Bose gas

We study the solitonic Lieb II branch of excitations the in one-dimensional Bose gas in homogeneous and trapped geometry. Using Bethe-ansatz Lieb's equations we calculate the ``effective number of atoms'' and the ``effective mass'' of the excitation. The equations of motion of the excitation are defined by the ratio of these quantities. The frequency of oscillations of the excitation in a harmonic trap is calculated. We show that the chemical potential exhibits a nonmonotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. We demonstrate that the quadratic coefficient in the expansion low-temperature expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity, while the quartic coefficient experiences opposite-direction effects originating from (a) non-linearity of the excitation spectrum (b) negative excluded-volume corrections. Finally, the energy of a 1D Bose gas is found in a box with zero boundary condition. It is shown that a standard solution dating back to Gauden's work in 1971 in the thermodynamic limit actually describes a dark soliton and not zero boundary condition.