Spontaneous symmetry breaking of solitons trapped in a double-channel potential

We consider a two-dimensional (2D) nonlinear Schrödinger equation with self-focusing nonlinearity and a quasi-1D double-channel potential, i.e., a straightforward 2D extension of the well-known double-well potential. The model may be realized in terms of nonlinear optics and Bose-Einstein condensates. The variational approximation (VA) predicts a bifurcation breaking the symmetry of 2D solitons trapped in the double channel, the bifurcation being of the subcritical type. The predictions of the VA are confirmed by numerical simulations. The work presents an original example of the spontaneous symmetry breaking of 2D solitons in dual-core systems.

Figure 1

The shape of the quasi-one-dimensional double-well potential $U(x,y)$.

Figure 2

(Color online) The subcritical symmetry-breaking bifurcation in the double-channel model, as predicted by the variational approximation through Eq. (15). Unstable branches of the solutions are shown by dashed curves.

Figure 3

(Color online) The supercritical symmetry-breaking bifurcation for the cw states, in the 1D model.

Figure 4

(Color online) Imbalance in the norm between half-planes $x>0$ and $x<0$, defined as per Eq. (18) for numerically found stationary soliton solutions, versus the total norm. The continuous and dashed lines show, respectively, numerically found stable symmetric and asymmetric solutions (unstable solutions were not generated by the numerical procedure). The parameters are $L=1$, $D=1$, and $U_0=1$.

Figure 5

(Color online) Examples of numerically found asymmetric (a) and symmetric (b) soliton solutions of Eq. (3), for $n=4.05$. Other parameters are the same as in Fig. 4. The gray shading indicates the position of the two potential wells. Panels (c) and (d) show the corresponding solutions as predicted by the variational approximation. In panels (e) and (f), the cross sections of the solutions along $y=0$ are compared: The dashed and solid curves correspond to the numerical and variational solutions, respectively.

Figure 6

(Color online) Typical examples of the evolution of perturbed stationary states produced by numerical simulations of Eq. (1). (a) A stable symmetric state for $n=3.9$. (b), (c) Stable asymmetric and unstable symmetric states for $n=4.15$. Other parameters are as in Fig. 4. The evolution time is $t=300$.

Figure 7

(Color online) Comparison of the variational prediction for the asymmetric solitons (solid line) with numerical results (dotted and dashed lines). The dashed curve corresponds to the same parameters as in Fig. 4, while the dotted one is generated by the numerical solution of Eq. (3) with the distance between the wells increased to $L=2$.