Optimal control of dark solitons

We consider optimal control methods for dark solitons and their higher-order states based on the completely integrable nonautonomous nonlinear Schrodinger equation (NLSE) models with varying gravitational-like and harmonic oscillator potentials, and in the presence of gain or absorption. In view of the fact that the time-dependence of the dark soliton phase (chirp) represents a major difference between bright and dark solitons, our main goal here is to gain a deeper understanding at a fundamental level based on the Lax pairs of how this time-dependent phase appears in the framework of the nonautonomous NLSE models with varying dispersion and nonlinearity. Specifically, we consider dynamics of dark solitons and their higher-order states in a variety of different optical scenarios. The main features of dark solitons are supplied by numerical calculations performed both for the exactly integrable and nonintegrable cases. We reveal nontrivial dynamics of black and gray solitons under the optimal control conditions. Since the experimental realization of dark solitons is possible only if a finite background pedestal is used, we consider dynamics of dark solitons and their higher-order states superimposed on the ground state of nonlinear harmonic oscillator. Direct numerical experiments performed in the framework of the Gross-Pitaevskii (GP) model in a broad range of parameters reveal many fundamental features of the dark solitons dynamics in confining harmonic oscillator potential, including the background BEC cloud spreading near the Feshbach resonance (when repulsive nonlinearity has a dispersive form). One of the most important effects is related to the saturation of the maximum value of self-compression near the Feshbach resonance. This effect exists both for the central black soliton and for the gray solitons formed during the higher-order dark soliton oscillations in the parabolic trap. The soliton becomes black at the turning points in the harmonic potential and gray at the trap center. This makes possible to observe the nontrivial dynamics of the periodic transformation of the black soliton to the gray one.