BOSE-EINSTEIN CONDENSATION (BEC): Part III. Dark solitons

The dynamics of dark solitons is studied within the framework of the mathematical model of nonlinear Schrodinger equation (NSE) with an external harmonic potential. A comparative analysis of the solution of no stationary problems is performed for a linear harmonic oscillator and the NSE model with a harmonic potential for different signs of the self-action potential. It is shown that the main specific feature of the dynamics of dark NSE solitons in a parabolic trap is the formation of solitons with dynamically changing form factors producing the periodic variation in the modulation depth (the degree of "blackness") of dark solitons. The oscillation period of the dark solitons does not coincide with the oscillation period of a linear quantum-mechanical oscillator, which is caused by the periodic transformation of the black solitons to the gray one and vice versa.