Three-Dimensional Solitary Waves and Vortices in a Discrete Nonlinear Schrödinger Lattice

In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schrödinger equation, we find discrete vortex solitons with various values of the topological charge $S$. Stability regions for the vortices with $S=0,1,3$ are investigated. The $S=2$ vortex is unstable and may spontaneously rearranging into a stable one with $S=3$. In a two-component extension of the model, we find a novel class of stable structures, consisting of vortices in the different components, perpendicularly oriented to each other. Self-localized states of the proposed types can be observed experimentally in Bose-Einstein condensates trapped in optical lattices and in photonic crystals built of microresonators.

Figure 1

The ILM with $S=0$ is shown for $C=1$. (a) shows the 3D contour plot $\mathrm{R}\mathrm{e}(u_{l,m,n})=0.5$. (b) shows 2D cross sections of the solution through $n=6$ (top) and $n=5$ (bottom) for the $11\times 11\times 11$ DNLS lattice.

Figure 2

(a),(b) show level contours at $\mathrm{R}\mathrm{e}(u_{l,m,n})=\pm 0.5$ and $\mathrm{I}\mathrm{m}(u_{l,m,n})=\pm 0.5$, respectively, for the 3D vortex ILM with $S=1$. Red or light-gray and blue or dark-gray surfaces pertain to the levels with $-0.5$ and $+0.5$ values, respectively. Cross sections of the vortex are shown in (c),(d). (e),(f) display the development of the instability of the vortex for $C=0.7$, through the time evolution of its amplitude, and a 2D cut of the profile at $t=100$ [(f) top and bottom, respectively]. The unstable vortex transforms itself into an ordinary ILM with $S=0$. (e) shows the spectral plane $({\lambda }_r,{\lambda }_i)$ of the linear stability eigenvalues for the same unstable vortex.

Figure 3

The ILM vortex with $S=2$ for $C=0.01$. (a),(b) show level contours at $\mathrm{R}\mathrm{e}(u_{l,m,n})=\pm 0.25$ and $\mathrm{I}\mathrm{m}(u_{l,m,n})=\pm 0.25$, respectively. (c) displays the linear stability eigenvalues, while (d) shows the result of the long evolution of this unstable vortex. The eventual state, shown through its 2D cross sections at $t=1000$, is a vortex with $S=3$ (see Fig. 4), which is stable for this value of $C$.

Figure 4

Stable stationary vortex ILMs with $S=3$ for $C=0.01$. (a),(b) have same meaning as (a),(b) in Fig. 3.

Figure 5

A complex of two orthogonal vortices with $S=1$ in the two-component system is shown for $C=0.01$. (a),(b) and (c),(d) correspond to the first and second components, respectively. All panels have the same meaning as (a),(b) in Fig. 3.