Vortex families near a spectral edge in the Gross-Pitaevskii equation with a two-dimensional periodic potential

We examine numerically vortex families near band edges of the Bloch wave spectrum for the Gross-Pitaevskii equation with two-dimensional periodic potentials and for the discrete nonlinear Schrödinger equation. We show that besides vortex families that terminate at a small distance from the band edges via fold bifurcations, there exist vortex families that are continued all the way to the band edges.

Figure 1

Family of vortex solutions of Eq. (7) continued from the vortex (6) via the envelope approximation (8) at $\omega =-0.03$ (point $D$). (a) A fixed computational domain is used with $N=85$. (b) Detail of the vicinity of the spectral edge for a range of sizes of the computational domain.

Figure 2

(a)–(g) Modulus of the discrete vortex solutions labeled A–G in Fig. 1. (h) Plots of the complex phases for vortices $A$ and $G$.

Figure 3

Family of vortex solutions of Eq. (1) continued from the vortex (6) via the envelope approximation (2) at $\omega \approx 4.09$ (point $B$). (a) A fixed computational domain is used with $L=93$. (b) Detail of the vicinity of the spectral edge for a range of sizes of the computational domain.

Figure 4

Modulus of the continuous vortex solutions labeled in Fig. 3. The inset in (a) shows a qualitative plot of the complex phase.

Figure 5

Family of discrete vortex solutions continued from a quadrupole vortex and four nearest neighbor excited sites as $\omega \to -\infty$.

Figure 6

Modulus of the discrete quadrupole vortex solutions labeled in Fig. 5. The inset in (a) shows a qualitative plot of the complex phase.

Figure 7

Family of discrete vortex solutions continued from a dipole vortex with two nearest neighbor excited sites as $\omega \to -\infty$.

Figure 8

Modulus of the discrete dipole vortex solutions labeled in Fig. 7.