Stationary through-flows in a Bose-Einstein condensate with a $\mathcal{PT}$-symmetric impurity

Superfluid currents in the boson condensate with a source and sink of particles are modeled by the $\mathcal{PT}$-symmetric Gross-Pitaevskii equation with a complex potential. We demonstrate the existence of through-flows of the condensate—stationary states with the asymptotically nonvanishing flux. The through-flows come in two broad varieties determined by the form of their number density distribution. One variety is described by diplike solutions featuring a localized density depression; the other one comprises humplike structures with a density spike in their core. We exemplify each class by exact closed-form solutions. For a fixed set of parameters of the $\mathcal{PT}$-symmetric potential, stationary through-flows form continuous families parametrized by the strength of the background flux. All humplike and some diplike members of the family are found to be stable. We show that the through-flows can be controlled by varying the gain-and-loss amplitude of the complex potential and that these amplitude variations may produce an anomalous response of the flux across the gain-loss interface.

Figure 1

The Wadati potentials (7) and (10) (top row) and the corresponding exact solutions (8) and (11) (bottom row). The left column (column I) corresponds to $a=-1/2,\varphi =\pi /6$; the middle column (II) to $a=-1/2,\varphi =\pi /3$; and the right column (III) to $a=1/2,\varphi =\pi /3$. The three columns exemplify potentials and solutions at points I, II, and III in Fig. 2. In the top row, the blue and red lines depict the real and imaginary parts of the potential (7) and (10). In the bottom row, the blue curves describe the number density and the red ones show the flux $J$ associated with the solution (8) and (11). In these plots, $\mu =2$.

Figure 2

The parameter plane for the Wadati potential (7) and (10) and the corresponding solution (8) and (11). In the regions I and II (colored blue) the solution has the form of a dip in a constant density—see panels (b.I) and (b.II) of Fig. 1. In the domain demarcated by III (tinted brown), the soliton is humplike—see panel (b.III) of Fig. 1. Note that the domain III does not have an upper bound and extends indefinitely in $a$. The red curve separating domains I and II is given by $a=-{sin}^2\varphi$. Along this curve and along the vertical line $\varphi =0$, the potential is real and the through-flow reduces to the nonlinear Schrödinger (NLS) dark soliton. Along the other boundary, $\varphi =\pi /2$, the potential is the $\mathcal{PT}$-symmetric Scarff II.

Figure 3

A family of through-flows supported by the $\mathcal{PT}$-symmetric Gaussian potential (20) with $\gamma =3/2$ and $U_0=9/4$. The left panel traces $n\left(0\right)$ as $J_{\infty }$ is varied, and the right panel displays the corresponding $J\left(0\right)$. Solid (black) and dashed (red) segments of curves correspond to stable and unstable solutions, respectively. The bottom panels show examples of a dip (c) and a hump (d) pertaining to $J_{\infty }=-1$. The blue line depicts $n\left(x\right)$ and the green one shows $J\left(x\right)$. In all panels, the background density is set to $n_{\infty }=1$; the largest value of $|J_{\infty }|$ attainable by this family is determined by Eq. (6): $J_{\infty }=-2$. The starting point of continuation is the top point with $J_{\infty }=0$; here $n\left(0\right)$ and $J\left(0\right)$ are determined by the exact solution (21): $n\left(0\right)=1$ and $J\left(0\right)=3$.

Figure 4

The evolution of a hump (a) and (b) and coexisting dip solution (c) and (d). The panels (a) and (c) display the density $n(x,t)={\left|\mathrm{\Psi }\right|}^2$, and the panels (b) and (d) show the flux $J(x,t)=2{(arg\mathrm{\Psi })}_x{\left|\mathrm{\Psi }\right|}^2$. The hump solution was initially perturbed by a Gaussian-shaped perturbation with an amplitude of $2\%$ of the amplitude of the hump. No explicit perturbation was added to the dip solution initially; here the instability was seeded by the discretization error. In these plots, $U_0=9/4,\gamma =0.4,n_{\infty }=1$, and $J_{\infty }=-0.8$.

Figure 5

The density at $x=0$ and the interfacial flux associated with the through-flows in the Gaussian $\mathcal{PT}$-symmetric potential (20) with $U_0=9/4$ and varying $\gamma$. Panels (a) and (b) illustrate the case $J_{\infty }=0$. Panels (c) and (d) correspond to $J_{\infty }=-0.8$. Solid lines trace stable through-flows; unstable solutions are marked by dashes. In this figure, $n_{\infty }=1$.