Symmetry breaking in linearly coupled dynamical lattices

We examine one- and two-dimensional models of linearly coupled lattices of the discrete-nonlinear-Schrödinger type. Analyzing ground states of the system with equal powers (norms) in the two components, we find a symmetry-breaking phenomenon beyond a critical value of the total power. Asymmetric states, with unequal powers in their components, emerge through a subcritical pitchfork bifurcation, which, for very weakly coupled lattices, changes into a supercritical one. We identify the stability of various solution branches. Dynamical manifestations of the symmetry breaking are studied by simulating the evolution of the unstable branches. The results present the first example of spontaneous symmetry breaking in two-dimensional lattice solitons. This feature has no counterpart in the continuum limit because of the collapse instability in the latter case.

Figure 1

The bifurcation diagram for the discrete solitons in the anticontinuum limit, $ϵ=0$; $r$ and $E$ are the asymmetry parameter and the half of the total squared norm, respectively (definitions are given in the text). The solid and dashed lines show stable and unstable solutions, respectively.

Figure 2

The bifurcation diagrams for $ϵ=1.6$ in the 1D model. The dashed-dotted line indicates solutions found by means of the variational approximation, while solid and dashed lines show, respectively, numerically found stable and unstable steady-state solutions.

Figure 3

The top panel shows the bifurcation diagram in the 2D model for $ϵ=0.25$, in the same way (i.e., with the same meaning of the different curves) as the 1D diagram is shown in Fig. 2. The bottom panel displays the dependence of the solution’s total squared norm $E$, upon the chemical potential $\mu$, for the symmetric solutions. Unlike the 1D case, there are two different symmetric solutions for many values of $E$, resulting in both stable (solid line) and unstable (dashed lines) solutions for norms below the value at which the symmetric and asymmetric solution branches intersect.

Figure 4

Plots of solutions belonging to different branches in Fig. 2, at $E=3.4$. The top-row figures show the solution profiles found by means of the numerical $\left(U_N,V_N\right)$ and variational (“analytical”, $U_A,V_A$) methods. The bottom-row plots illustrate the stability eigenvalues for the numerical solution. The first column presents a stable stationary asymmetric solution belonging to the outer (upper) branch in Fig. 2, the second column is an unstable asymmetric solution from the inner branch, and the last column is taken from the stable part of the family of symmetric solutions, with $r=0$.

Figure 5

Time-evolution plots for perturbed one-dimensional solutions from Fig. 2. The top (middle) row depicts the evolution of ${\left|U\right|}^2$ $\left({\left|V\right|}^2\right)$ by means of space-time contour plots. The first two columns pertain to two different perturbations of the solution shown in the second column of Fig. 4: the first column is generated by a perturbation which pushes the solution toward the upper stable asymmetric solution branch in the bifurcation diagram, while, in the second column, the initial perturbation pushes the solution toward the $r=0$ solution. The last column pertains to a perturbation of the unstable symmetric solution (belonging to the branch with $r=0$) at $E=4.1$, which pushes the solution toward the upper stable branch. The bottom row displays the value of asymmetry parameter $r$ as the solutions evolve in time (solid line), and the constant value of $r$ for the steady states at the same value of the norm (dashed horizontal lines).

Figure 6

Cross-section plots of the asymmetric solutions belonging to different branches in Fig. 3, at $E=1.435$. The top-row figures show the solutions found by means of the numerical $\left(U_N,V_N\right)$ and variational (“analytical”, $U_A,V_A$) methods, and the bottom-row plots display stability eigenvalues for the numerical solution. As can be seen, the first and second columns represent, respectively, stable and unstable solutions belonging to the upper (outer) and inner curves (asymmetric branches) of the bifurcation diagram, respectively.

Figure 7

Same as Fig. 6 for two symmetric solutions found at $E=1.435$.

Figure 8

(Color online) Time-evolution plots for a perturbed unstable solution in Fig. 6. The top-row perturbation pushes the solution toward the upper stable part of the curve, while the bottom-row perturbation pushes it to the symmetric branch. The left column shows the three-dimensional space-time evolution of isodensity contours of the $U$ (respective top subpanels) and $V$ (respective bottom subpanels) modes, while the right column shows the evolution of the asymmetry measure $r$, from an unstable steady state to a stable one (both are denoted by dashed lines).

Figure 9

(Color online) Similar to the previous figure: the time-evolution plots for two perturbed unstable symmetric states. The top row pertains to the solution with $E=1.5$, and the bottom row is for the solution from Fig. 7. In the top row, the perturbation pushes the solution toward the upper asymmetric branch, while in bottom row the perturbation initiates the evolution of the solution toward the stable symmetric state.