Busch-Anglin effect for matter-wave and optical dark solitons in external potentials

Discovered by Busch and Anglin, the most important and puzzling nonlinear effect for solitons in external potentials lies in the fact that in a harmonic oscillator potential, a dark soliton oscillates with a frequency that is smaller by a factor of 2 in comparison with the ground state oscillations frequency. We review the main features of the Busch-Anglin effect and show that the physical nature of the self-defocusing nonlinearity can radically alter the Busch-Anglin dynamics. To bridge the gap between the matter-wave and optical dark solitons, we consider both the impact of linear and nonlinear absorption and the crucial role of nonlocality (or nonstationarity) of the nonlinear response in the triggering of the Busch-Anglin effect. We conclude that nonlocality (or nonstationarity) brings into existence the self-induced Busch-Anglin effect. Both linear and nonlinear losses cannot radically affect the period of dark solitons oscillations. We establish that from the physical point of view, it is best to redefine the specific dark soliton angle between the asymptotic phases of non-vanishing boundary conditions in accordance with Faddeev and Takhtajan's mathematical arrangement based on the phase shift across a dark soliton “core” controlling the soliton “blackness”. Among other things, we show that the effects analogous to the self-induced Busch-Anglin effect exist for bright solitons as well. Namely, triggered by nonlocal (or time-delayed) self-focusing nonlinearity, the bright solitons oscillate in the parabolic trap with a shifted equilibrium position and a frequency that exactly coincides with the oscillator frequency.