Spontaneous symmetry breaking of fundamental states, vortices, and dipoles in two- and one-dimensional linearly coupled traps with cubic self-attraction

We introduce two- and one-dimensional (2D and 1D) systems of two linearly coupled Gross-Pitaevskii equations (GPEs) with the cubic self-attraction and harmonic-oscillator (HO) trapping potential in each GPE. The system models a Bose-Einstein condensate with a negative scattering length, loaded in a double-pancake trap, combined with the in-plane HO potential. In addition to that, the 1D version applies to the light transmission in a dual-core waveguide with the Kerr nonlinearity and in-core confinement represented by the HO potential. The subject of the analysis is spontaneous symmetry breaking in 2D and 1D ground-state (GS, alias fundamental) modes, as well as in 2D vortices and 1D dipole modes. (The latter ones do not exist without the HO potential.) By means of the variational approximation and numerical analysis, it is found that both the 2D and 1D systems give rise to a symmetry-breaking bifurcation (SBB) of the supercritical type. The stability of symmetric and asymmetric states, produced by the SBB, is analyzed through the computation of eigenvalues for perturbation modes and verified by direct simulations. The asymmetric GSs are always stable, while the stability region for vortices shrinks and eventually disappears with the increase of the linear-coupling constant, $\kappa$. The SBB in the 2D system does not occur if $\kappa$ is too large (at $\kappa >{\kappa }_{max}$); in that case, the two-component system behaves, essentially, as its single-component counterpart. In the 1D system, both asymmetric and symmetric dipole modes feature an additional oscillatory instability, unrelated to the symmetry breaking. This instability occurs in several regions which expand with the increase of $\kappa$.

Figure 1

(a) The cross section of the 2D profile of a stable symmetric ground state with $(\kappa ,N)=(0.4,1)$. (b, c) The same for a stable asymmetric ground state with $(\kappa ,N)=(0.4,4)$. Red solid and black dashed curves designate numerical and VA results, respectively.

Figure 2

Total norm $N$ versus chemical potential $\mu$ for (a) symmetric (“SY”) and (b) asymmetric (“ASY”) ground states (modes with $S=0$) in the 2D system, for indicated values of coupling constant $\kappa$. Red and black curves represent the numerical and variational results, respectively, see Eqs. (27) and (30). The dashed segment of the numerically generated $N\left(\mu \right)$ curve in (a) represents the branch destabilized by the SBB. (c) Critical values of total norm $N_{\mathrm{cr}}^{(S=0)}$ at the symmetry-breaking point of the GS in the 2D system versus $\kappa$. (d) Critical values of the respective chemical potential, ${\mu }_{\mathrm{cr}}^{(S=0)}$, versus $\kappa$. Again, red and black curves represent, severally, numerical and VA results, see Eq. (32). The numerically found dependences displayed in (c) and (d) terminate at the point determined by Eq. (60).

Figure 3

(a, b) The numerically simulated evolution of an unstable 2D symmetric fundamental ($S=0$) state (shown is cross-section $y=0$ ) with $(\kappa ,N)=(0.2,3)$, which demonstrates the onset of the spontaneous symmetry breaking with concomitant oscillations. (c, d) The respective evolution of densities of the two components at the center.

Figure 4

(a, b) The evolution of an unstable symmetric fundamental soliton (shown is cross section $y=0$) with $(\kappa ,N)=(0.2,11.5)$, which demonstrates the acceleration of the symmetry breaking in the course of the development of the collapse. In panel (c), this is additionally illustrated by the dependence of asymmetry $\theta$ between the two components [see Eq. (10)] on time for the same solution.

Figure 5

Bifurcation diagrams, in the $(N,\theta )$ plane, for the fundamental modes (ground states) with $S=0$, at different values of the linear-coupling constant: (a) $\kappa =0.2$, (b) $\kappa =0.4$, and (c) $\kappa =0.6$. Here, black and continuous (dashed) red curves represent the variational results, and numerically found stable (unstable) solutions, respectively.

Figure 6

(a) A stable symmetric vortex with $S=1$ and $(\kappa ,N)=(0.4,8)$. (b, c) Two components of a stable asymmetric vortex with $(\kappa ,N)=(0.4,8.8)$.

Figure 7

Total norm $N$ versus chemical potential $\mu$ for symmetric and asymmetric vortices with $S=1$, at indicated values of the coupling constant: (a) $\kappa =0.1$, (b) $\kappa =0.15$, and (c) $\kappa =0.2$. The decoupled system with $\kappa =0$ is included in (a), too, for completeness’ sake. Inset (d) is a zoom of a small stability region which exists in (c). Solid and dashed lines designate stable and unstable branches, respectively. Arrows indicate points where the asymmetric vortices loose their stability. They are $N_{max}^{(S=1)}$ for the asymmetric ones, see Eq. (64).

Figure 8

The critical value of the total norm at the symmetry-breaking point $N_{\mathrm{cr}}^{(S=1)}$ versus the coupling constant $\kappa$ for vortices with $S=1$ corresponds to the black line with circles. The chemical potential at the bifurcation point ${\mu }_{\mathrm{cr}}^{(S=1)}$ versus the coupling constant $\kappa$ is shown by the black line with triangles. As indicated in the figure, these dependences can be fitted by linear relations (62). They terminate at the point given by Eq. (63).

Figure 9

Bifurcation diagrams, in the $(N,\theta )$ plane, for the vortices with $S=1$ at different values of the linear-coupling constant: (a) $\kappa =0.1$, (b) $\kappa =0.15$, and (c) $\kappa =0.2$. Stable and unstable branches are shown by solid and dashed lines, respectively.

Figure 10

The evolution of an unstable 2D symmetric ($S=1$) vortex (shown is cross-section $y=0$) with $(\kappa ,N)=(0.4,8.8)$, which demonstrates the onset of the spontaneous symmetry breaking with residual oscillations.

Figure 11

Numerically simulated evolution of an unstable symmetric vortex with $S=1$ and $(\kappa ,N)=(0.8,18)$, initiated by random noise (at the amplitude level of $5\%$) added to the input. This is a typical example of the splitting instability.

Figure 12

Numerically simulated evolution of an unstable asymmetric vortex with $S=1$ and $(\kappa ,N)=(0.2,6)$, initiated by random noise (at the amplitude level of $5\%$) added to the input. This figure presents a typical example of the crescent instability (see the main text).

Figure 13

The stability diagram for symmetric (a) and asymmetric (b) vortices with $S=1$ in the plane of $(\kappa ,N)$. The red and gray colors designate, respectively, stability and instability areas. In panel (a), the small blue area represents instability with residual oscillations, while the gray area below the dashed-dotted curve designates the crescent instability similar to that shown in Fig. 12. Above the dotted line, this instability leads to the collapse of the component with the larger norm, but the collapse does not happen in the small gray subarea below the dotted line. The gray area above the dashed-dotted and dashed lines represents the splitting instability scenario displayed in Fig. 11, while robust recurrent splitting-recombination cycles take place in the gray area below the dashed curve. In panel (b), the gray area under the dashed-dotted line represents the instability shown in Fig. 12, while in the area between the dashed and dashed-dotted lines instability similar to that in Fig. 12 occurs, finally leading to the collapse. Finally, the gray area above the dashed curve represents instability similar to that displayed in Fig. 11.

Figure 14

(a, b) Typical examples of stable symmetric solutions for the 1D ground-state and dipole modes, with $(\kappa ,N)=(0.4,1)$. Red solid and black dashed curves display the numerical and variational results, respectively. (c, d) The numerically simulated onset of the symmetry breaking in an unstable symmetric ground-state mode, with $(\kappa ,N)=(0.4,2.5)$. Panels (e) and (f) display several examples of stable asymmetric ground-state solutions, with $\left(\kappa ,N\right)=(0.4,2)$, and dipole modes, with $\left(\kappa ,N\right)=(0.4,2.7)$. Red and green solid curves represent two different components, as produced by the numerical solution, while black dashed lines depict their VA-predicted counterparts.

Figure 15

Total norm $N$ versus chemical potential $\mu$ for symmetric (“SY”) (a) and asymmetric (“ASY”) (b) 1D ground state at $\kappa =0.4$. Red and black curves represent the numerical and variational results, the latter given by Eqs. (41) and (44) for the symmetric and asymmetric states, respectively. (c, d) The same for the 1D dipole states, with the variational predictions given by Eqs. (52) and (55). Dashed are unstable segments of the numerically generated branches.

Figure 16

Bifurcation diagrams, in the $(N,\theta )$ plane, for the 1D ground-state modes, at different values of the linear-coupling constant: (a) $\kappa =0.2$, (b) $\kappa =0.4$, (c) $\kappa =0.6$, and (d) $\kappa =0.8$. The red and black curves represent numerical findings and VA predictions, respectively.

Figure 17

The same as in Fig. 16, but for the 1D dipole modes.

Figure 18

(a) Critical values of total norm $N_{\mathrm{cr}}$ at which the 1D ground state undergoes the symmetry-breaking transition versus coupling coefficient $\kappa$. (b) The same for the 1D dipole states. Again, red and black curves represent numerical and variational results, respectively [see Eqs. (46) and (58), with regard to the VA predictions].

Figure 19

The evolution of an unstable asymmetric dipole mode, with $(\kappa ,N)=(0.6,4)$, which transforms into a spatially confined turbulent state.