Symmetry-breaking instability of leapfrogging vortex rings in a Bose-Einstein condensate

Three coaxial quantized vortex rings in a Bose-Einstein condensate exhibit aperiodic leapfrogging dynamics. It is found that such circular vortex rings are dynamically unstable against deformation breaking axial rotational symmetry. The dynamics of the system is analyzed using the Gross-Pitaevskii and vortex-filament models. The dependence of the instability on the initial arrangement of the vortex rings is investigated. The system is found to be significantly unstable for a specific configuration of the three vortex rings.

Figure 1

(a)–(c) Dynamics of vortex rings obtained by solving the GP equation with $g=1$. The isodensity surface of the density ${\left|\psi \right|}^2=0.5$ is shown. (a) Two vortex rings initially located with $R_1=R_2=20$, $Z_1=-28$, and $Z_2=-26$. (b) Three vortex rings with a triangular initial arrangement, where $R_1=20-2\sqrt{3}/3$, $R_2=R_3=20+1/\sqrt{3}$, $Z_1=-26$, $Z_2=-28$, and $Z_3=-24$. (c) Three vortex rings with a rectilinear initial arrangement, where $R_1=R_2=R_3=20$, $Z_1=-27$, $Z_2=-26$, and $Z_3=-28$. The vortices seen from the $+z$ direction at $t=160$ are highlighted. Axial symmetry is retained in (a) and (b) and broken in (c) in the time evolution. The size of the box is ${64}^3$, with the origin at the center. See the Supplemental Material for movies showing the dynamics [30]. (d) Initial vortex arrangements set by Eq. (5). The triangular and rectilinear arrangements used in (b) and (c) are shown in the left-hand and right-hand panels, respectively, where the closed circles represent the positions of vortex cores on the plane including the $z$ axis.

Figure 2

Dependence of the symmetry-breaking dynamics on the initial distance $d$ between the vortex rings in the rectilinear initial arrangement with radius $R=20$. The dynamics is obtained by solving the GP equation with $g=1$ and (a) $d=1$, (b) $d=2$, and (c) $d=4$. The dynamics in (b) is the same as that in Fig. 1. The isodensity surface of the density ${\left|\psi \right|}^2=0.2$ is shown. The size of the box is ${64}^3$, with the origin at the center, and is seen from the $+z$ direction. See the Supplemental Material for movies showing the dynamics [30].

Figure 3

Dependence of the symmetry-breaking dynamics on the interaction coefficient $g$ for vortex rings in a rectilinear initial arrangement with $d=2$, $R=20$, and (a) $g=2$ and (b) $g=0.5$. The dynamics is obtained by solving the GP equation. The isodensity surfaces of the densities ${\left|\psi \right|}^2=0.5$ and 0.2 are shown for (a) and (b), respectively. The size of the box is ${64}^3$, with the origin at the center, and is seen from the $+z$ direction. See the Supplemental Material for movies showing the dynamics [30].

Figure 4

Dynamics of vortex rings obtained by the vortex-filament model. (a) Two vortex rings initially located with $R_1=R_2=20$, $Z_1=-30$, and $Z_2=-26$. (b) Three vortex rings with a triangular initial arrangement, where $R_1=20-4\sqrt{3}/3$, $R_2=R_3=20+2/\sqrt{3}$, $Z_1=-28$, $Z_2=-26$, and $Z_3=-30$. (c) Three vortex rings with a rectilinear initial arrangement, where $R_1=R_2=R_3=20$, $Z_1=-30$, $Z_2=-26$, and $Z_3=-22$. Vortices are seen from the $+z$ direction in the rightmost panels. Axial symmetry is retained in (a) and (b) and broken in (c). The size of the box is ${64}^3$, with the origin at the center. See the Supplemental Material for movies showing the dynamics [30].

Figure 5

Time evolution of the Fourier components $C_m$ defined in Eq. (13) for (a) triangular and (b) rectilinear initial arrangements, corresponding to the dynamics in Figs. 4 and 4, respectively. The vertical dashed lines in (b) indicate $t=40$ and $t=75$, for which $C_m$ is exponentially rising. The insets in (b) show the vortex rings at these times.

Figure 6

The most unstable eigenvector of Eq. (14) for $m=4$. The three vortex rings are initially in the rectilinear configuration, as in Fig. 4. (a) The most unstable eigenvectors $(c_4^{\left(1\right)},d_4^{\left(1\right)})$, $(c_4^{\left(2\right)},d_4^{\left(2\right)})$, and $(c_4^{\left(3\right)},d_4^{\left(3\right)})$ are plotted at each time, where the circles, triangles, and squares correspond to the vortex rings in the inset. (b) Shape of the most unstable mode with $m=4$ at $t=40$ and $t=75$. The three vortex rings are seen from the $+x$ direction in the left-hand panel at each time. In the right-hand panels, the three vortex rings seen from the $+z$ direction are separately shown for clarity. Despite the order of the vortex rings being changed by leapfrogging, the shapes of the modes at $t=40$ and $t=75$ are similar to each other.

Figure 7

Dependence of the symmetry-breaking dynamics on initial distance $d$ between vortex rings for the rectilinear initial arrangement with radius $R=20$. The dynamics is obtained by the vortex-filament model for (a) $d=6$ with $Z_1=-70$, $Z_2=-64$, and $Z_2=-58$ and (b) $d=3$ with $Z_1=-30$, $Z_2=-27$, and $Z_2=-24$. The size of the box is ${64}^3$, with the $z$ axis at the center; $-80<z<-16$ is shown at $t=0$ in (a) and $-32<z<32$ is shown in the other boxes. The rightmost panels show the system seen from the $+z$ direction. See the Supplemental Material for movies showing the dynamics [30].