Vector azimuthons in two-component Bose-Einstein condensates

We introduce matter-wave vector azimuthons, i.e., spatially localized vortex states with azimuthal modulations of density, in multicomponent Bose-Einstein condensates. These localized states generalize spatially modulated vortex solitons introduced earlier in nonlinear optics [A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005)] and Bose-Einstein condensates [V. M. Lashkin, Phys. Rev. A 77, 025602 (2008)]. We find, numerically, nonrotating and rotating two-component azimuthons in a Bose-Einstein condensate with a negative scattering length confined by a quasi-two-dimensional parabolic trap.

Figure 1

Localized soliton-azimuthon solutions of Eqs. (2) with topological indices $l_1=0$ and $l_2=1$ for ${\mu }_1=-2$ $\left(N_1=10.3\right)$ and ${\mu }_2=3$ $\left(N_2=4.3\right)$. The differences between the two cases (a) and (b) are defined by the angular velocity $\omega$ and the modulation parameter $p_2$, namely, (a) $\omega =0.13$, $p_2=0.28$, and (b) $\omega =0.23$, $p_2=0.72$. All quantities are in dimensionless units.

Figure 2

(Color online) Family of soliton-azimuthons with $l_1=0$ and $l_2=1$ shown as dependencies of the numbers of particles $N_{1,2}$ in two components and the total number of particles $N=N_1+N_2$ on the chemical potential ${\mu }_1$ for (a) ${\mu }_2=1$ and $\omega =1$ and (b) ${\mu }_2=3$ and $\omega =0.2$. All quantities are in dimensionless units.

Figure 3

(Color online) Top row: vector soliton-azimuthon with $l_1=0$ and $l_2=2$. Other parameters are ${\mu }_1=-2$ $\left(N_1=12.2\right)$ and ${\mu }_2=1$ $\left(N_2=30.7\right)$, angular velocity $\omega =0.6$, and modulation parameter $p_2=0.24$. Bottom row: phase structure of the corresponding components. All quantities are in dimensionless units.

Figure 4

(Color online) Dependence of the angular velocity $\omega$ on the modulation parameter $p_2$ for ${\mu }_1=-2$ and ${\mu }_2=1$ (squares), ${\mu }_2=2$ (circles), and ${\mu }_2=3$ (triangles) for two families of coupled soliton-azimuthon pairs (a) $\left(l_1,l_2\right)=\left(0,1\right)$ and (b) $\left(l_1,l_2\right)=\left(0,2\right)$. All quantities are in dimensionless units.

Figure 5

(Color online) Stability of the soliton-azimuthon pair with ${\mu }_1=-2$ and ${\mu }_2=0.5$. (a) Unstable dynamics for azimuthon modulation parameter $p_2=0.6$, angular velocity $\omega =1.4$, and the period of rotation $T=2\pi /\omega =4.5$. (b) Stable spiraling with $p_2=0.84$, $\omega =1.6$, and $T=3.9$. All quantities are in dimensionless units.

Figure 6

(Color online) Stable dynamics of the soliton-azimuthon pair with ${\mu }_1=-2$ and ${\mu }_2=3$. (a) $\omega =0$ and $p_2=0$. (b) $\omega =0.23$ and $p_2=0.72$. All quantities are in dimensionless units.

Figure 7

Azimuthon-azimuthon vector states with $\left(l_1,l_2\right)=\left(+1,+1\right)$ in (a) and $\left(l_1,l_2\right)=\left(+1,+2\right)$ in (b). Parameters of the displayed solutions are (a) angular velocity $\omega =0.9$ with ${\mu }_1=1.7$, $p_1=0.42$ $\left(N_1=5.6\right)$ and ${\mu }_2=1$, $p_2=0.3$ $\left(N_2=13.7\right)$; (b) $\omega =0.055$ with ${\mu }_1=4.8$, $p_1=0.18$ $\left(N_1=4.3\right)$ and ${\mu }_2=4.8$, $p_2=0.12$ $\left(N_2=10.2\right)$. All quantities are in dimensionless units.