Lieb Mode in a Quasi-One-Dimensional Bose-Einstein Condensate of Atoms

We calculate the dispersion relation associated with a solitary wave in a quasi-one-dimensional Bose-Einstein condensate of atoms confined in a harmonic, cylindrical trap in the limit of weak and strong interactions. In both cases, the dispersion relation is linear for long-wavelength excitations and terminates at the point where the group velocity vanishes. We also calculate the dispersion relation of sound waves in both limits of weak and strong coupling.

Figure 1

(a) Dispersion relation $\mathcal{E}=\mathcal{E}(\mathcal{P})$ in the limit of weak interactions for a solitary wave (solid curve) and for the Bogoliubov mode (dotted line.) The energy is measured in units of ${\mathcal{E}}_0$ and the momentum in units of ${\mathcal{P}}_0$. (b) Same as (a) in the limit of strong interactions, with the energy measured in units of ${\mathcal{E}}_{\mathcal{0}}^{\mathcal{\prime }}$. The inset shows the same graph on a smaller scale, for long-wavelength excitations.