Solitons and vortices in two-dimensional discrete nonlinear Schrödinger systems with spatially modulated nonlinearity

We consider a two-dimensional (2D) generalization of a recently proposed model [Gligorić et al., Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual “extended” unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being those considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (also analytically, when possible). Typical scenarios of instability development are exhibited through direct simulations.

Figure 1

Numerically generated dependencies $N\left(\epsilon \right)$ for the families of single-site, in-phase two-site, and out-of-phase two-site modes are shown in the left, middle, and right panels, respectively.

Figure 2

The branch corresponding to the single-site excitation. The left panel shows eigenvalues bifurcating from the edge of the continuous spectrum band (the first around $\varepsilon =0.055$ and two more at a slightly larger value of $\varepsilon )$, rapidly approaching the spectral-plane's origin as $\varepsilon \to 0.082$, the value at which the present branch terminates. Here and in similar plots displayed below, the dependence of the edge of the continuous wave band on $\varepsilon ,\lambda =\pm (1-8\varepsilon )i$, is shown by the red dashed line. The right panel shows the profile of the branch at $\varepsilon =0.08$.

Figure 3

In-phase configuration: the left panel shows the imaginary eigenvalue growing from the origin, as per numerical results (blue solid line), according to the analytical prediction (green dash-dotted line), and in the homogeneous model (the lower dash-dotted line, also obtained in an analytical form). Eigenvalues bifurcating from the edge of the continuous-spectrum band are also shown by blue solid lines. The right panel displays an example of this wave form for $\varepsilon =0.079$.

Figure 4

Similar to the previous graph but for the out-of-phase configuration. In this case, however, the eigenvalue pair bifurcating from the origin moves to the real line and hence the real part of the relevant eigenvalue is shown (blue solid line: numerical linear stability result; green dash-dotted line: theory). The right panel shows the corresponding wave form for $\varepsilon =0.07$.

Figure 5

Four-site, in-phase excitation: In the left panel of the figure three pairs of imaginary eigenvalues bifurcating from the origin are shown by the blue solid line (the lower parabolic line corresponds to a double pair). The corresponding analytical prediction is shown by the green dash-dotted line, while the prediction for the homogeneous model is presented by the magenta dash-dotted line. The edge of the continuous-spectrum band is shown by the red dashed line. The right panel shows the configuration for $\varepsilon =0.075$.

Figure 6

Similar to the previous figures, but now for the four-site, out-of-phase excitation. The left panel shows the numerical result (the blue solid line) and the analytical prediction (the green dash-dotted line) for the three pairs bifurcating from the origin towards the real axis, rendering the configuration highly unstable. A typical example of the configuration profile for $\varepsilon =0.055$ is shown in the right panel.

Figure 7

The left panels of the figure illustrate the real (top) and imaginary (bottom) parts of the spatial distribution of a two-dimensional vortex cross. The phases of the four excited sites are 0, $\pi /2,\pi$, and $3\pi /2$, so that a phase circulation of $2\pi$ is achieved when moving along a contour surrounding the mode's pivot. The solution is shown for $\varepsilon =0.068$, and its corresponding spectral plane is displayed in the bottom right panel. The top right panel illustrates the $O\left(\varepsilon \right)$ (or weaker; see the smallest eigenvalue pair) dependence for small $\varepsilon$ of the eigenvalue pairs bifurcating from the origin. It is the collision of these pairs with the ones bifurcating from the band edge (which is depicted, as before, by the dashed red line), that leads to the instability at $\varepsilon \ge 0.068$.

Figure 8

Same as the above figure, but now for a vortex square in the case of a nonlinearity profile symmetric around $(0.5,0.5)$ in the form of Eq. (8). The top left panel (real part) and bottom left panel (imaginary part of the solution), as well as the spectral plane of the bottom right panel are given for $\varepsilon =0.074$. The top right panel illustrates the trajectory of the imaginary part of the eigenvalues associated with the stability of the configuration up to $\varepsilon \approx 0.08$.

Figure 9

The left panel of the figure shows the dependence of the amplitude of the extended solution on $\varepsilon$, while the right panel displays a typical profile of the extended mode for $\varepsilon =0.2$.

Figure 10

Evolution of the unstable two-site, in-phase mode at $\varepsilon =0.079$. The top left panel shows the final profile of the absolute value of the discrete wave form at the final simulation time of $t=600$, while the bottom left panel shows the difference between absolute values of the top left profile and the initial one. The right panels show the absolute value at the central site (top) and at two adjacent ones (bottom); the blue solid line corresponds to the initially excited $(1,0)$ site, and the red dashed line to the $(0,1)$ site.

Figure 11

Similar to the previous figure but now for the out-of-phase two-site configuration at $\varepsilon =0.07$. The top left panel shows the result of the evolution at $t=600$, while the bottom left panel shows its difference from the input, in terms of the absolute value. The right panels illustrate the evolution at the central (top) and nearest-neighbor $(1,0)$ (bottom) sites, demonstrating the exponential nature of the instability.

Figure 12

The same as previous figures, but now for the four-site, in-phase excited state with $\varepsilon =0.075$. Notice the broadening of the solution (left panels) and the oscillatory manifestation of the instability (right panels).

Figure 13

The same as previous figures, but for the exponential instability dominating the dynamics of an out-of-phase configuration at $\varepsilon =0.055$.

Figure 14

The same as previous figures, but for the vortex with $\varepsilon =0.068$. The top right panel shows populating the originally empty central site, while the four sites surrounding it no longer carry equal amplitudes, in the course of the development of the oscillatory instability (the bottom right panel).