Topological solitons and Bloch oscillations in Bose-Einstein condensate

We studied the evolution of topological solitons in the space of its width d and the conjugated momenta l during Bloch oscillations. The unstable solitons are confined in a well with its boundary as four branches of a hyperbola, the stable solitons sit at the four branches of the hyperbola (d · l = 28) which is in agreement with the generalized Heisenberg uncertainty principal Δd · Δl ? Const. The generation, annihilation, and bifurcation of solitons is going on in the well. In studying the behavior of the nonlinear interaction parameter for the stable solitons, it is found that bright solitons and dark solitons alternatively come into action during the breakdown and revival of Bloch oscillations.