Soliton oscillations in collisionally inhomogeneous attractive Bose-Einstein condensates

We investigate bright matter-wave solitons in the presence of a spatially varying nonlinearity. It is demonstrated that a translation mode is excited due to the spatial inhomogeneity and its frequency is derived analytically and also studied numerically. Both cases of purely one-dimensional and “cigar-shaped” condensates are studied by means of different mean-field models, and the oscillation frequencies of the pertinent solitons are found and compared with the results obtained by the linear stability analysis. Numerical results are shown to be in very good agreement with the corresponding analytical predictions.

Figure 1

(Color online) (a) The linear spectrum of a bright soliton of the GP equation (1) for $ϵ=0.01$, $N=1.2$, and $\mu =-0.18$. The bright soliton’s translation mode has an eigenfrequency $\mid \Omega \mid =0.049$ in excellent agreement with the prediction of Eq. (10). (b) Solid line shows the translation mode’s frequency as a function of $ϵ$ obtained by the BdG equations (13, 14), as well as the theoretical prediction of Eq. (10) for $N=2.8$ and $\mu =-1$; the two results are identical and the respective curves cannot be distinguished from each other. Stars depict the soliton oscillation frequency obtained by the numerical integration of Eq. (1).

Figure 2

(Color online) Solid line shows the density of a bright matter-wave soliton with $N=1.2$ and $\mu =-0.18$ (the parameter values are the ones used in Fig. 1); the soliton is initially placed at $z_0=2$. Dashed line shows the nonlinear coefficient $g\left(z\right)$ for $ϵ=0.01$. The initial displacement of the soliton, $z_0$, which sets the amplitude of the soliton oscillation, is such that the inhomogeneous nonlinearity remains attractive in the region where the soliton oscillation takes place.

Figure 3

(Color online) (a) Similar to panel (a) of Fig. 1, but for the NPSE model of Eq. (3). The inhomogeneity parameter is the same $(ϵ=0.01)$ and the chemical potential is chosen $(\mu =0.762)$ so that the norm of the bright soliton is the same in both GP and NPSE models. The translation mode’s frequency is up-shifted, i.e., $\mid \Omega \mid =0.062$, as compared to the respective value calculated with the 1D GP equation (depicted by stars). (b) The translation mode’s frequency $\Omega$ as a function of the norm $N$, calculated by the BdG analysis of the NPSE model (dashed line) and the 1D GP model (solid line). The eigenfrequency $\Omega$ is clearly upshifted due to increase of the dimensionality. The inhomogeneity parameter is again $ϵ=0.01$.

Figure 4

(Color online) Spatiotemporal contour plot of the soliton’s longitudinal density for $N=0.757$, $\mu =0.92$, and initial soliton position $z_0=2$. The dashed line shows the harmonic oscillation of the soliton center, with a frequency predicted by the BdG analysis of the NPSE model. Note that for the chosen value of $ϵ$ the inhomogeneous nonlinearity remains attractive for $-10<x<10$.