Stable stationary and quasiperiodic discrete vortex breathers with topological charge $S=2$

We demonstrate the stability of a stationary vortex breather with vorticity $S=2$ in the two-dimensional discrete nonlinear Schrödinger model for a square lattice and also discuss the effects of exciting internal sites in a vortex ring. We also point out the fundamental difficulties of observing these solutions with current experimental techniques. Instead, we argue that relevant initial conditions will lead to the formation of quasiperiodic vortex breathers.

Figure 1

(Color online) (a) $\mid {\Psi }_{n,m}\mid$ and (b) $\arg \left\{{\Psi }_{n,m}\right\}$ for a stable stationary vortex of topological charge $S=2$ for coupling $C=0.1$ (mode $i$ in Table ).

Figure 2

The radial dependence of the initial condition (5) for two different values of the parameters: $S=1$, $A=2$, $w=1$, $r_0=0$ (solid line) and $S=2$, $A=0.5$, $w=5$, $r_0=1.5$ (dotted line). The squares (◻) indicate the values at sites along a lattice direction and the circles (●) values along a direction diagonally in the lattice for unit lattice spacing.

Figure 3

(Color online) The dynamics of the initial condition with $S=2$ in Fig. 2 against a noisy background for $C=0.049$. (a) $\mid {\Psi }_{n,m}(t=0)\mid$ and (b) $\arg \left\{{\Psi }_{n,m}(t=0)\right\}$, where only the phase of the main excited sites are shown for clarity. As the vortex evolves, without much change to the initial amplitudes, the vorticity will change from $S=2$ in (b) to $S=0$ in (c) at $t=2.6$ and to $S=-2$ in (d) at $t=5.3$ and back again. The quasiperiodicity is also clearly seen in plot (e) of the real parts of the amplitudes at site $(-2,0)$ (solid line) and site $(-1,1)$ (dotted line), but as can be inferred from plot (f) of $\mid {\Psi }_{-2,0}\mid$ it is not an exact two-frequency quasiperiodic solution. The calculations were made on a $21\times 21$ lattice with periodic boundary conditions.

Figure 4

(a) The distribution of Floquet multipliers for an exact quasiperiodic vortex breather with ${\Lambda }_0=0.3450$ and ${\Lambda }_b=0.5950$ for $C=0.049$ showing its linear stability. The system size is $11\times 11$. (b) The time evolution of $\mid {\Psi }_{-2,0}\mid$ for the same solution illustrating the periodicity of the oscillations with period $T_b∕2=\pi ∕{\Lambda }_b=5.2800$ as is expected for an exact solution. Compare this with Fig. 3f and note also that the magnitude of the oscillations differ by about three orders of magnitude.