Dark solitons as quasiparticles in trapped condensates

We present a theory of dark soliton dynamics in trapped quasi-one-dimensional Bose-Einstein condensates, which is based on the local-density approximation. The approach is applicable for arbitrary polynomial nonlinearities of the mean-field equation governing the system as well as to arbitrary polynomial traps. In particular, we derive a general formula for the frequency of the soliton oscillations in confining potentials. A special attention is dedicated to the study of the soliton dynamics in adiabatically varying traps. It is shown that the dependence of the amplitude of oscillations vs the trap frequency (strength) is given by the scaling law $X_0\propto {\omega }^{-\gamma }$ where the exponent $\gamma$ depends on the type of the two-body interactions, on the exponent of the polynomial confining potential, on the density of the condensate, and on the initial soliton velocity. Analytical results obtained within the framework of the local-density approximation are compared with the direct numerical simulations of the dynamics, showing a remarkable match. Various limiting cases are addressed. In particular for the slow solitons we computed a general formula for the effective mass and for the frequency of oscillations.

Figure 1

Time dependence of the soliton coordinate on the adiabatically changing frequency, modeled by the function $\omega \left(t\right)=(1+0.001t)\times 0.1$, in logarithmic coordinates: panels (a)–(c) and (e)–(g). Straight solid lines visualize the law $X_0={\tilde{X}}_0{\omega }^{-\gamma }$. Dashed lines in panels (a) and (e) show the exponents ${\gamma }_{\pm }$ [see Eq. (17)] while in the rest of the panels only the average $\gamma$ is shown. In panels (a)–(c) the parameters are $n_0=1$ and $v=0.1$, $0.5$, and $0.8$, correspondingly. The respective matching parameter is ${\tilde{X}}_0=0.385$, $1.93$, $2.6$. In panels (e)–(g) the parameters are $v=0.1$, $n_0=0.3;0.6$, and $1$, correspondingly. The respective matching parameter is ${\tilde{X}}_0=0.622$, 0.445, 0.385. In panel (d) we plot dependence of $\gamma$ on the initial soliton velocity $v$ for the case $n_0=1$. In panel (h) we depict the dependence of $\gamma$ on the initial density $n_0$ corresponding to the case of the initial velocity $v=0.1$. In the numerical calculations we take $\hslash =1$, $m=1$, and $g=1$.

Figure 2

Dependence of the soliton coordinate on the frequency in the logarithmic scale. Adiabatic change of the frequency is modeled by the law $\omega \left(t\right)=(1+0.001t)\times 0.1$. The dashed lines visualize the law $X_0={\tilde{X}}_0{\omega }^{-\gamma }$. In panels (a)–(c) the parameters are $n_0=1$ and $v=0.1$, 0.3, and $0.6$, respectively. The matching parameter is ${\tilde{X}}_0=0.49$, 0.98, 1.52. In panels (e)–(g) the parameters are $v=0.1$, and $n=0.4$, 0.7, and $1$, respectively. The matching parameter is ${\tilde{X}}_0=0.38$, 0.445, 0.49. In panel (d) we show the dependences of the exponent $\gamma$ vs soliton velocity $v$ at the density $n_0=1$. In panel (h) we show the dependences of the exponent $\gamma$ vs density $n_0$ at $v=0.1$. In the numerical calculations we used $\hslash =1$, $m=1$, and $g=1$.

Figure 3

(a) Dependence of the position of the center of the QNLS dark soliton on time for $\omega =0.1$, $n_0=1$, and $v=0.1$ (b) Time dependence of the half period $T∕2$ substracted from (a). As before we take $\hslash =1$, $m=1$, and $g=1$.

Figure 4

Dependence of the soliton coordinate on the frequency adiabatically varying according to the law $\omega \left(t\right)=(1+0.001t)\times 0.11$. Straight lines show the law $X_0\propto {\omega }^{-\gamma }$. In panels (a) and (d) we show also the laws $X_{0,\pm }\propto {\omega }^{-{\gamma }_{\pm }}$ [see Eq. (17)] by dashed lines. In (a)–(c) parameters are $n_0=1$, and $v=0.14$, 0.42, and $0.85$, correspondingly. In (d)–(f) parameters are $v=0.14$, and $n_0=0.4$, 0.6, and $1$, correspondingly. In the numerical calculations we take $\hslash =1$, $m=1$, and $g=1$.