Lagrangian approach to the dynamics of dark matter-wave solitons

We analyze the dynamics of dark matter-wave solitons on a Thomas-Fermi cloud described by the Gross-Pitaevskii equation with radial symmetry. One-dimensional, ring, and spherical dark solitons are considered, and the evolution of their amplitudes, velocities, and centers is investigated by means of a Lagrangian approach. In the case of large-amplitude oscillations, higher-order corrections to the corresponding equations of motion for the soliton characteristics are shown to be important in order to accurately describe its dynamics. The numerical results are found to be in very good agreement with the analytical predictions.

Figure 1

Motion of the center of a dark soliton in a quasi-1D BEC confined in a parabolic trap with $\Omega =0.035$ and TF radius $\approx 40$. The soliton is initially placed at $x_0\left(0\right)=0$, and its initial velocity is $A\left(0\right)=0.7$. The almost coinciding solid and dash-dotted lines correspond, respectively, to the results of the direct numerical integration of the GPE and the analytical predictions of Eqs. (13, 14). The dashed and dotted lines are results obtained from simplified versions of Eqs. (13, 14), corresponding to the leading- [Eq. (15) for $D=1$] and first-order approximations in Eq. (6) (see text).

Figure 2

Same as in Fig. 1, but for a cylindrical (ring) dark soliton with initial velocity $A\left(0\right)=0$. The initial value of the ring radius $R$ is $R\left(0\right)=18.2\equiv R_{\min }$, while $R_{\max }=22.2$. Note that at $t=110$, the snaking instability sets in.