Bose–Einstein condensation in an external potential at zero temperature: Solitary-wave theory

For a trapped, dilute atomic gas of short-range, repulsive interactions at extremely low temperatures, when Bose–Einstein condensation is nearly complete, some special forms of the time-dependent condensate wave function and the pair-excitation function, the latter being responsible for phonon creation, are investigated. Specifically, ~i! a class of external potentials Ve(r,t) that allow for localized, shapepreserving solutions to the nonlinear Schrödinger equation for the condensate wave function, each recognized as a solitary wave moving along an arbitrary trajectory, is derived and analyzed in any number of space dimensions; and ~ii! for any such external potential and condensate wave function, the nonlinear integro-differential equation for the pair-excitation function is shown to admit solutions of the same nature. Approximate analytical results are presented for a sufficiently slowly varying trapping potential. Numerical results are obtained for the condensate wave function when Ve is a time-independent, spherically symmetric harmonic potential. © 1999 American Institute of Physics. @S0022-2488~99!03211-9#