Discrete solitons and vortices on anisotropic lattices

We consider the effects of anisotropy on solitons of various types in two-dimensional nonlinear lattices, using the discrete nonlinear Schrödinger equation as a paradigm model. For fundamental solitons, we develop a variational approximation that predicts that broad quasicontinuum solitons are unstable, while their strongly anisotropic counterparts are stable. By means of numerical methods, it is found that, in the general case, the fundamental solitons and simplest on-site-centered vortex solitons (“vortex crosses”) feature enhanced or reduced stability areas, depending on the strength of the anisotropy. More surprising is the effect of anisotropy on the so-called “super-symmetric” intersite-centered vortices (“vortex squares”), with the topological charge $S$ equal to the square’s size $M$: we predict in an analytical form by means of the Lyapunov-Schmidt theory, and confirm by numerical results, that arbitrarily weak anisotropy results in dramatic changes in the stability and dynamics in comparison with the degenerate, in this case, isotropic, limit.

Figure 1

(a) Norm of the solution vs $\Lambda$ for several values of the anisotropy and fixed coupling strength $ϵ=1$. For all the panels in this figure, the anisotropy values are $\alpha =1.5$, 1.25, 1, and 0.75, respectively, for each curve from top to bottom. Thick lines (solid and dashed) represent direct numerical results and thin lines represent the VA. The dashed lines correspond to unstable soliton solutions. Note that the sign of the slope of $N\left(\Lambda \right)$ reflects the stability of the soliton solutions as predicted by the VK criterion. (b) The norm of the soliton solution as a function of the coupling strength for fixed $\Lambda =1$. Once again, thick lines represent direct numerical results and thin lines illustrate the VA.

Figure 2

(Color online) The line in the top panel shows the critical value of $ϵ$ (the border between stable and unstable discrete solitons) as a function of $\alpha$; the dashed line is $ϵ=1∕\sqrt{\alpha }$. The middle panel shows how the real and imaginary parts of the stability eigenvalue, ${\lambda }_r$ and ${\lambda }_i$, depend on $ϵ$ for $\alpha =1.25$, 1, and 0.75 (dashed, solid, and dash-dotted curves, respectively). The bottom panel shows an example of the discrete soliton found for $ϵ=1$ and $\alpha =1.5$.

Figure 3

(Color online) The top panel shows the critical value of $ϵ$ separating the stable and unstable discrete vortices (on-site-centered ones, alias vortex crosses) with $S=1$ as a function of $\alpha$. The middle panel shows how the real and imaginary parts of the eigenvalue leading to the instability depend on $ϵ$ for $\alpha =1.3$, 1, and 0.7 (dashed, solid, and dash-dotted curves, respectively). Notice that, for $\alpha =1.3$, there is a secondary instability arising for $ϵ>0.455$. The bottom panel shows the squared-absolute-value profile of the discrete vortex for $ϵ=0.5$ and $\alpha =0.2$.

Figure 4

For the $S=1$ supersymmetric square vortex (one with the size $M=1$), two real eigenvalues are shown as functions of $ϵ$ for $\alpha =1.05$. The numerical and analytical results (see text) are displayed, respectively, by the solid lines and dashed lines.

Figure 5

The dynamics of an initially very weakly perturbed supersymmetric vortex with $S=M=1$, principally based on four lattice sites that form an elementary cell (the sites are labeled as 1,2,3,4). The time evolution of the squared absolute value of the fields at these sites is shown in the top panel for a weakly anisotropic model, with $\alpha =1.05$, and for its isotropic counterpart $(\alpha =1)$ in the bottom panel. In both cases, the same uniformly distributed, random initial perturbation of amplitude ${10}^{-4}$ was added to the solution at $t=0$ to excite possible instabilities. Clearly, the vortex on the weakly anisotropic lattice becomes unstable at $t>50$, while in the isotropic case the perturbation remains bounded and small at all times. In these examples, the intersite lattice coupling constant is $ϵ=0.025$.

Figure 6

For the $S=M=2$ supersymmetric vortex, the three real eigenvalues are displayed as functions of $ϵ$ for $\alpha =1.05$. The solid and dashed lines depict the numerical and analytical results.

Figure 7

Same as in Fig. 5 but for the supersymmetric vortex of the $S=M=2$ type. The different lines depict the squared absolute values of the field at the eight sites carrying the vortex in the anisotropic (top) model and its isotropic counterpart (bottom) for $ϵ=0.015$.