We study the solitonic Lieb II branch of excitations in one-dimensional Bose-gas in homogeneous and trapped geometry. Using Bethe-ansatz Liebs 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. It changes continuously from its soliton-like value omega_h/sqrt{2} in the high density mean field regime to omega_h in the low density Tonks-Girardeau regime with omega_h the frequency of the harmonic trapping. Particular attention is paid to the effective mass of a soliton with velocity near the speed of sound.
A version of the Greens functions theory of the Van der Waals forces which can be conveniently used in the presence of spatial dispersion is presented. The theory is based on the fluctuation-dissipation theorem and is valid for interacting bodies, separated by vacuum. Objections against theories acounting for the spatial dispersion are discussed.
It is shown that criticism of my paper arXiv:0801.0656 Phys. Rev. Lett, vol. 101, 163202 (2008) by the authors of Comment arXiv:0810.3243v1 is wrong and that their main arguments are in contradiction with established concepts of statistical physics.
A new theory describing the interaction between atoms and a conductor with small densities of current carriers is presented. The theory takes into account the penetration of the static component of the thermally fluctuating field in the conductor and generalizes the Lifshitz theory in the presence of a spatial dispersion. The equation obtained for the force describes the continuous crossover between the Lifshitz results for dielectrics and metals.
