Management of matter-wave solitons in Bose-Einstein condensates with time-dependent atomic scattering length in a time-dependent parabolic complex potential

In this paper, we consider a Gross-Pitaevskii (GP) equation with a time-dependent nonlinearity and a spatiotemporal complex linear term which describes the dynamics of matter-wave solitons in Bose-Einstein condensates (BECs) with time-dependent interatomic interactions in a parabolic potential in the presence of feeding or loss of atoms. We establish the integrability conditions under which analytical solutions describing the modulational instability and the propagation of both bright and dark solitary waves on a continuous wave background are obtained. The obtained integrability conditions also appear as the conditions under which the solitary waves of the BECs can be managed by controlling the functional gain or loss parameter. For specific BECs, the dynamics of bright and dark solitons are investigated analytically through the found exact solutions of the GP equation. Our results show that under the integrability conditions, the gain or loss parameter of the GP equation can be used to manage the motion of both bright and dark solitons. We show that for BECs with loss (gain) of atoms, the bright and dark solitons during their propagation have a compression (broadening) in their width. Furthermore, under a safe range of parameters and under the integrability conditions, it is possible to squeeze a bright soliton of BECs with loss of atoms into the assumed peak matter density, which can provide an experimental tool for investigating the range of validity of the 1D GP equation. Our results also reveal that under the conditions of the solitary wave management, neither the injection or the ejection of atoms from the condensate affects the soliton peak during its propagation.

Figure 1

The evolution of modulational instability given by solution Eq. (15) with the parameters $A_c=A_s=1$, $k_c=0.03$, $k_s=0.04$, ${\alpha }_0=-1$, $\left|{\eta }_0\right|={\ell }_0=1$. (a, c) $g\left(t\right)=exp\left[2\lambda t\right]$, $\gamma =-\frac{\lambda }{2}$, $k=\frac{{\lambda }^2}{2{\alpha }_0}$, $\lambda =0.05$, $g_0=1$. (b, d) $g\left(t\right)=1+m_{0}sin\left[\omega t\right]$, $\gamma \left(t\right)=-\frac{m_0\omega }{4}\frac{cos\left[\omega t\right]}{1+m_{0}sin\left[\omega t\right]}$, $k\left(t\right)=\frac{m_0{\omega }^2}{8{\alpha }_0}\left(\frac{2m_0+2sin\left[m_{0}t\right]+m_0{cos}^2\left[\omega t\right]}{{\left(1+m_{0}sin\left[\omega t\right]\right)}^2}\right)$, $\omega =1,m_0=0.4$.

Figure 2

(a, b) Spatial profiles at time $t=5$, (c, d) the evolution plots, and (e, f) the contour plots of the density ${\left|\psi (x,t)\right|}^2$ for the case when $\alpha$ and $\beta$ in solution Eq. (9b) satisfy condition $\alpha \beta \ne 0.$ The data used to generate plots (a), (c), and (e) and plots (b), (d), and (f) are the same as for plots (a) and (b) in Fig. 1, respectively. Here, we used $\rho =0.02$ with a positive $\beta$.

Figure 3

The dynamics of a bright solitary wave in the expulsive parabolic potential given by Eq. (16). Different plots are generated with parameters $A_c=1.12$, $A_s=3.2$, $\left|g_0\right|=0.25$, $k_c=0.03$, $k_s=0.04$, and $\lambda =0.02$.

Figure 4

The evolution plots of the maximal intensity ${\left|\psi \right|}_{max}^2$ (dotted lines), the minimal intensity ${\left|\psi \right|}_{min}^2$ (dashed lines), and the background intensity ${\left|{\psi }_c\right|}^2$ (solid lines) for BECs with loss of atoms. To generate these plots, we have used the parameters $A_c=0.8$, $A_s=2.4,g_0=0.25$, $k_c=0.03$, $k_s=0.04$, $T_0=0$, and $\lambda =0.02$.

Figure 5

The atomic exchange between the bright solitons and the background given by Eq. (14d) for BECs with $g\left(t\right)=2\left|g_0\right|exp\left[\lambda t\right]$ and $k\left(t\right)=-\frac{1}{4}{\lambda }^2$. The range of time is (a) $t\in \left[0,400\right]$, (b) $t\in \left[0,60\right]$, and (d) $t\in \left[100,130\right]$. Plot (a) corresponds to a zero cw background, while plots (b) and (c) are obtained are associated with a nonzero cw background. The parameters are given as follows: $A_c=1.12$, $A_s=3.2,g_0=0.25,k_c=0.03,k_s=0.04$, and $\lambda =0.02$.

Figure 6

The evolution plots of ${\left|\psi (x,t)\right|}^2$ associated with the dark soliton described by Eq. (17)) with sign “$-$.” (a) The profile of dark soliton at different times: $t=0$ (solid line), $t=0.2$ (dashed line), $t=0.4$ (dotted line), and $t=0.6$ (dot-dashed line); (b) the evolution contour plot of density ${\left|\psi (x,t)\right|}^2$ of dark soliton. The horizontal (red) line in panel (a) shows the density of the cw background. The parameters used to generate these plots are $g_0=0.25$, $A_c=1.12$, $A_s=3.2$, $k_c=0.03$, $k_s=0.04$, and $\lambda =2$.

Figure 7

Plots of different scenarios for a BEC with the temporal periodic modulation of the $s$-wave scattering length [17] with the nonlinearity parameter $g\left(t\right)=1+m_{0}sin\left[\omega t\right]$, $0<m_0<1$. (a) Time evolution of the gain/loss parameter $\gamma \left(t\right)=-\frac{m_0\omega }{4}\frac{cos\left[\omega t\right]}{1+m_{0}sin\left[\omega t\right]}$; (b) the evolution plot of the density ${\left|\psi (x,t)\right|}^2$ of the bright soliton described by Eq. (13); (c) plot of the total number of atoms $N_0\left(t\right)$ defined by Eq. (19a); (d) the evolution contour plot of the density ${\left|\psi (x,t)\right|}^2$; (e) time-periodic atomic exchange between the bright soliton and the background given by Eq. (14d); (f) the evolution plots of ${\left|\psi \right|}_{max}^2$ (solid line) and ${\left|\psi \right|}_{min}^2$ (dashed line) described by Eq. (14a). The horizontal (red) line is the background intensities ${\left|{\psi }_c\right|}^2=A_c^2$. The range of time is two periods, i.e., $t\in \left[0,4\pi /\omega \right]$. The parameters used in these plots are $A_c=1$, $A_s=5.2$, $k_c=0.03$, $k_s=0.04$, $T_0=0$, $\omega =1$, and $m_0=0.2$.

Figure 8

The evolution of bright solitary waves on cw background given by Eq. (20) with $t_0=40$ for parameters $A_c=1$, $A_s=5.2$, $g_0=0.25$, $k_c=0.03$, and $k_s=0.04$. Plots (a) and (b) show respectively the evolution and the contour plots of density ${\left|\psi (x,t)\right|}^2$; plots (d) and (e) show the time-aperiodic atomic exchange between the bright soliton and the cw background in the range of time $t\in \left[0,20\right]$ and $t\in \left[30,50\right]$, respectively. Panel (c) plots the atomic exchange in the situation of zero background.

Figure 9

The evolution plot (a) and the contour plot (b) of density ${\left|\psi (x,t)\right|}^2$ of the bright soliton described by Eq. (22) with positive $\beta$ and $t_0=-40$. (c, d) The time-aperiodic atomic exchange $\chi \left(t\right)$ between the bright solitons and the cw background. (c) shows the number of atoms in the absence of the cw background, i.e., when $A_c=0$. The parameters used in these plots are $A_c=1$, $A_s=5.2$, $k_c=0.03$, $g_0=0.25$, and $k_s=0.04$.

Figure 10

The dynamics (a) and the evolution plot (b) of the bright soliton described by Eq. (22) with positive $\beta$ and $t_0=40$. (c–e) The time-aperiodic atomic exchange $\chi \left(t\right)$ between the bright solitons and the cw background. The range of time is (c) $t\in \left[0,200\right]$ for the case of zero cw background, (d) $t\in \left[0,30\right]$ and $t\in \left[50,300\right]$. The parameters used in these plots are the same as in Fig. 9.

Figure 11

The evolution (a) and the contour (b) plots of the density ${\left|\psi (x,t)\right|}^2$ of the dark soliton embedded on a cw background described by Eq. (23) with sign “$+$” for $t_0=2$. The parameters used in these plots are $A_c=1$, $A_s=5.2$, $k_c=0.03$, $k_s=0.04$, and $g_0=-0.25$.