Surface solitons in three dimensions

We study localized modes on the surface of a three-dimensional dynamical lattice. The stability of these structures on the surface is investigated and compared to that in the bulk of the lattice. Typically, the surface makes the stability region larger, an extreme example of that being the three-site “horseshoe”-shaped structure, which is always unstable in the bulk, while at the surface it is stable near the anticontinuum limit. We also examine effects of the surface on lattice vortices. For the vortex placed parallel to the surface, the increased stability-region feature is also observed, while the vortex cannot exist in a state normal to the surface. More sophisticated localized dynamical structures, such as five-site horseshoes and pyramids, are also considered.

Figure 1

(Color online) Results for the dipoles oriented parallel and normal to the surface. (a) Norm $N$ versus the lattice coupling constant $\epsilon$. (b) Imaginary part of the linear stability eigenvalue: solid (blue) and dashed (black) lines correspond, respectively, to numerically found and analytically predicted forms. (c) Real part of the critical (in)stability eigenvalue: the dashed (red) and solid (blue) lines depict the normal- and parallel-oriented dipoles, respectively, while the dashed-dotted (green) line corresponds to the bulk dipole. (d) (In)stability eigenvalue for the parallel surface dipole placed at distances from the surface starting from zero and up to five lattice periods away (curves right to left). (e), (g) Configurations and (f), (h) corresponding spectral stability planes just above the instability threshold. The level contours, corresponding to $\mathrm{Re}\left(u_{l,n,m}\right)=\pm 0.5\max \left\{u_{l,n,m}\right\}$ are shown, respectively, in dark gray (blue) and gray (red). The instability thresholds for the dipoles oriented parallel and normal to the surface are, respectively, $\epsilon =0.117$ and 0.120. For comparison, the threshold for the bulk dipole is $\epsilon =0.114$.

Figure 2

(Color online) Stability of the three-site horseshoe. Panels are similar to those in Fig. 1. (c) compares the critical stability eigenvalue, as a function of the lattice coupling $\epsilon$, for the surface and bulk horseshoes [solid (blue) and dashed-dotted (green) lines, respectively]. The bulk horseshoe is always unstable (due to a purely real, higher-order eigenvalue), while the corresponding surface configurations have a stability region (the corresponding eigenvalue becomes imaginary in this case). (d) and (e) correspond to the surface horseshoe just above the stability threshold of $\epsilon =0.239$.

Figure 3

(Color online) Stability for the five-site horseshoe at the surface. Panels are identical to those in Fig. 2. In this case, the stability threshold is at $\epsilon =0.211$, while for the bulk five-site horseshoe it is $\epsilon =0.205$. (d) and (e) depict the configuration and the corresponding linear stability spectrum just above the critical point of $\epsilon =0.211$.

Figure 4

(Color online) Stability of quadrupole modes. The layout is similar to that in Fig. 3. In (c), due to the close proximity of the thresholds, a close-up is shown for the critical stability eigenvalue versus the lattice coupling constant $\epsilon$ for the parallel and normal surface modes, and the bulk one [solid (blue) and dashed (red) lines, and the dashed-dotted (green) line, respectively]. The threshold for the bulk mode is $\epsilon =0.068$, while for the normal and parallel quadrupoles it is, respectively, $\epsilon =0.070$ and 0.071. As before, (d) and (e) show the configuration just above the instability threshold along with its corresponding spectral-stability plane.

Figure 5

(Color online) Stability of the four-site vortex in a grid of size $11\times 11\times 11$. The dashed-dotted and solid lines show the bulk vortex and the one parallel to the surface, respectively. The layout is similar to that of the above figures. Instability in the bulk occurs at $\epsilon =0.438$, and in the parallel surface vortex at $\epsilon =0.505$. The vortex cannot exist with the orientation normal to the surface. Panels (d) and (e) show the parallel surface vortex just above the instability threshold of $\epsilon =0.485$. As in the previous figures, the level contours corresponding to $\mathrm{Re}\left(u_{l,n,m}\right)=\pm 0.5\max \left\{u_{l,n,m}\right\}$ are shown, respectively, in dark gray (blue) and gray (red), while the complementary level contours, defined as $\mathrm{Im}\left(u_{l,n,m}\right)=\pm 0.5\max \left\{u_{l,n,m}\right\}$, are shown by light gray (green) and very light gray (yellow) hues, respectively.

Figure 6

(Color online) Instability of pyramid-shaped structures. This configuration abuts on the base in the form of a rhombus, and includes the out-of-plane site with zero phase. Three variants of this configuration are displayed in (d)–(f): bulk and normal and parallel to the surface, respectively. The stability of the three different variants of the pyramid is essentially identical, all three of them being unstable.

Figure 7

(Color online) Low-amplitude modes in a finite grid of size $9\times 9\times 9$ with $\Lambda =1.0$. (a) Norm $N$ versus coupling constant $\epsilon$ for several modes whose low-amplitude limit is parametrized by vectors $\mathbf{m}$, as calculated numerically and predicted by approximation (15) (solid and dashed lines, respectively). For comparison, the dashed-dotted line depicts the norm for surface normal dipole. (b) Real part of the critical (in)stability eigenvalue, calculated numerically.

Figure 8

(Color online) Evolution of unstable dipoles: (a) a bulk dipole; (b) and (c) dipoles placed parallel and normal to the surface, respectively. In all cases, the dipole is subject to an oscillatory instability, which is responsible for the eventual concentration of most of the norm at a single site (i.e., the transition to a monopole). Parameters are $\Lambda =1$, $\epsilon =0.2$, the lattice has a size of $13\times 13\times 13$, and times are indicated in the panels. All isocontour plots are defined as $\mathrm{Re}\left(u_{l,n,m}\right)=\pm 0.75=\mathrm{Im}\left(u_{l,n,m}\right)$, and the initial configurations were perturbed with random noise of amplitude 0.01. The coding for the isocontours is as follows: dark gray (blue) and gray (red) colors pertain to isocontours of the real part of the solutions, while the light gray (green) and very light gray (yellow) colors correspond to the isocontours of the imaginary part.

Figure 9

(Color online) Evolution of the unstable three-site horseshoes: (a) bulk three-site horseshoe and (b) horseshoe oriented normally to the surface. In both cases, the unstable horseshoe is subject to an oscillatory instability, which leads to the eventual concentration of most of the norm in a single-site structure. The isocontours and parameters are the same as in Fig. 8 except that $\epsilon =0.3$.

Figure 10

(Color online) Evolution of unstable five-site horseshoes: (a) the bulk horseshoe, and (b) the five-site horseshoe oriented normal to the surface. In both cases, the unstable horseshoe is subject to an oscillatory instability, which triggers the transition to a monopole. The isocontours and parameters are the same as in Fig. 9.

Figure 11

(Color online) Evolution of unstable vortices: (a) the bulk vortex for $\epsilon =0.3$ and (b) the vortex parallel to the surface, for $\epsilon =0.6$ and $\Lambda =1$. The isocontour plots are defined by $\mathrm{Re}\left(u_{l,n,m}\right)=\pm 1=\mathrm{Im}\left(u_{l,n,m}\right)$.

Figure 12

(Color online) Evolution of unstable pyramids. Panels (a), (b), and (c) display, respectively, the transformation of a bulk pyramid, and of pyramids oriented normal and parallel to the surface, for $\epsilon =0.2$.