Multidimensional discrete compactons in nonlinear Schrödinger lattices with strong nonlinearity management

The existence of multidimensional lattice compactons in the discrete nonlinear Schrödinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. By averaging over the period of the fast modulations, an effective averaged dynamical equation arises with coupling constants involving Bessel functions of the first and zeroth kinds. We show that these terms allow one to solve, at this averaged level, for exact discrete compacton solution configurations in the corresponding stationary equation. We focus on seven types of compacton solutions. Single-site and vortex solutions are found to be always stable in the parametric regimes we examined. Other solutions such as double-site in- and out-of-phase, four-site symmetric and antisymmetric, and a five-site compacton solution are found to have regions of stability and instability in two-dimensional parametric planes, involving variations of the strength of the coupling and of the nonlinearity. We also explore the time evolution of the solutions and compare the dynamics according to the averaged equations with those of the original dynamical system. The possible observation of compactons in Bose-Einstein condensates loaded in a deep two-dimensional optical lattice with interactions modulated periodically in time is also discussed.

Figure 1

We plot typical examples of exact 2D compacton solutions of Eq. (7). In the top left panel we show a single-site solution, in the bottom left we show a double-site in-phase solution, and in the bottom right we show a double-site out-of-phase solution. In these three plots colors show the (real) values of $A_{n,m}$, where blue represents $A_{n,m}=A_0$ and red $A_{n,m}=-A_0$. No color represents zero amplitude. In the top right plot we show a vortex solution where the four central colored sites all have amplitude $A_0$ and $\text{arg}\left(v_{n,m}\right)\in \{-\pi /2,0,\pi /2,\pi \}$ is plotted by the four respective colors: dark red, light red, light blue, and dark blue. Here the length and width of the 2D grid are each $L=7$.

Figure 2

The plots show three additional examples of exact 2D compacton solutions of Eq. (7) on four and five lattice sites. In the left and middle panels symmetric and antisymmetric four-site compactons are shown, respectively, while the right panel depicts a five-site compacton. Dark blue and red colors are as in Fig. 1. The light blue color at the center of the rightmost plot represents the value $y$ for the center site of the five-site solution obtained from Eq. (16).

Figure 3

The plots show the graphical solution of Eq. (16) written as $G\left(y\right)=0$ (left panel) and the section at $m=0$ of a five-site compacton solution (right panel) of Eq. (7) for parameter values $\kappa =0.5,\alpha =1$, and ${\gamma }_0=2$ with amplitudes at the edge sites fixed in correspondence with the fifth zero $z_5$ of the Bessel function $J_0$, e.g., $A_0=\sqrt{z_5/\alpha }\approx 3.864$. The amplitude of the central site is taken as the zero of $G\left(y\right)$ at $y\approx 3.8695$ (black dot in the left panel), while the corresponding chemical potential is obtained from Eq. (15) as $\mu \approx -30.678$.

Figure 4

We plot $max\left[\mathrm{real}\right(\nu \left)\right]$ in order to show the strength of the instabilities and regions of stability. Parameter values common across all four plots are $\epsilon =0.01$ and $\mathrm{\Omega }={\gamma }_1=10$ so that $\alpha =1$. The left column of plots is associated with a double-site in-phase solution with lattice length both horizontally and vertically $L=17$. The right column of plots is associated with a double-site out-of-phase solution with lattice length $L=15$. The top row corresponds to $j=1$ and the bottom to $j=2$. Notice that the top right plot shows more striations in the coloring than the top left plot; these further smooth out with higher lattice size $L$. The green dots in the top two plots are parameter values for which the unstable solutions are propagated in time (see Figs. 7 and 8 below).

Figure 5

The panels show eigenvalues in the complex plane for selected parameters corresponding to a horizontal cut on the top left panel of Fig. 4 (in-phase solution), obtained by setting ${\gamma }_0=1.75$. For $\kappa =0$ (not shown) the eigenvalues are purely imaginary and four eigenvalues are zero. For small nonzero $\kappa$ there are only two zero eigenvalues and the other two previously zero eigenvalues have begun to increase in magnitude on the real axis; this is shown for $\kappa =0.01$ (red) and $\kappa =0.3$ (blue) in the top left panel. As $\kappa$ increases the solution stabilizes and these real eigenvalues move back towards zero horizontally along the real axis. For even higher $\kappa$ values (at which the solution remains stable), the two eigenvalues increase in magnitude again from zero, but this time along the imaginary axis; this is shown for $\kappa =0.4$ (red) and $\kappa =0.5$ (blue) in the top right panel. These eigenvalues merge with the band of eigenvalues on the imaginary axis while the (continuous spectrum) band lengthens; this is shown for $\kappa =1$ in the bottom left panel. At even higher $\kappa$ values, two complex conjugate purely imaginary eigenvalues emerge from the band at a magnitude higher than any eigenvalue in the band; this is depicted in the bottom right panel for $\kappa =5$. The eigenvalues in the bottom right panel that appear to have a very small nonzero real part are seen to actually have a zero real part when the eigenvalues are computed for a much higher lattice length $L$.

Figure 6

The figure is similar to Fig. 5, but here we select parameter values according to a horizontal cut on the top right panel of Fig. 4 (out-of-phase solution) by setting ${\gamma }_0=3$. For $\kappa =0$ (not shown) the eigenvalues are purely imaginary and four eigenvalues are zero. For small nonzero $\kappa$ there are only two zero eigenvalues and the other two previously zero eigenvalues have begun to increase in magnitude on the imaginary axis; this is shown for $\kappa =0.2$ (red) and $\kappa =0.35$ (blue) in the top left panel. After the two merge with the (continuous spectrum) band of eigenvalues on the imaginary axis a quartet of complex eigenvalues then emerges for higher $\kappa$; this is depicted for $\kappa =0.445$ (red) and $\kappa =0.45$ (blue) in the top right panel. For higher $\kappa$ values the four complex eigenvalues move in the complex plane, first increasing in real and imaginary parts and then decreasing in the real part until they merge again with the imaginary axis (meanwhile the continuous spectrum band on the imaginary axis lengthens); the bottom two plots show this restabilization with $\kappa =1.3$ (red) and $\kappa =1.38$ (blue) in the bottom left panel and $\kappa =1.7$ in the bottom right panel. At the higher $\kappa$ values (at which the solution remains stable) two complex conjugate purely imaginary eigenvalues emerge from the band at a magnitude higher than any eigenvalue in the band.

Figure 7

The plots show the result of propagation of the double-site in-phase exact compacton solution in the time parameter $t$ according to each of the equations (1) and (5). The values of $\epsilon ,\mathrm{\Omega },{\gamma }_1$, and $\alpha$ are the same as in Fig. 4. In the bottom left panel the plot shows ${log}_{10}\left(\right|u_{n,m}\left(0\right)\left|\right)$ plotted as a function of $n,m$, where $u_{n,m}\left(0\right)=v_{n,m}\left(0\right)$ is the initial double-site in-phase compacton configuration at $t=0$; this plot, of course, shows an order 0 logarithmic amplitude at the two excited sites and vanishing amplitude elsewhere. The instability strength for this solution is $max\left[\mathrm{real}\right(\nu \left)\right]\approx 0.5327$. In the bottom right panel the plot shows ${log}_{10}\left[\right|v_{n,m}\left(80\right)\left|\right]$, where $v_{n,m}\left(t\right)$ is determined from $v_{n,m}\left(0\right)$ according to the averaged equation (5); the plot of ${log}_{10}\left[\right|u_{n,m}\left(80\right)\left|\right]$ determined from the nonautonomous equation (1) is visually indistinguishable from the averaged version shown in the bottom right panel. In the bottom right panel plot we see small-amplitude excitations appearing near the compacton site and these small amplitudes continue to spread out spatially as $t$ increases, while the central sites have amplitudes that approach nonzero values. The top left panel shows this evolution of the amplitude of the two excited sites over time with the unaveraged magnitudes $|w_{0,0}\left(t\right)|=|u_{0,0}\left(t\right)|$ in blue and $|w_{1,0}\left(t\right)|=|u_{1,0}\left(t\right)|$ in green; these are propagated according to (1). The overlying red line plots correspond to the magnitudes $|w_{0,0}\left(t\right)|=|v_{0,0}\left(t\right)|$ and $|w_{1,0}\left(t\right)|=|v_{1,0}\left(t\right)|$ propagated according to the averaged equation (5). From the perspective displayed in the top left panel the averaged and the unaveraged amplitudes seem to lie on top of one another. The top right panel here shows a small portion of the same plot zoomed in so as to show how the averaged solution in red in fact averages the nonautonomous equation solution (over a period of the nonlinearity prefactor variation) in blue.

Figure 8

The plots are similar to Fig. 7, with the same parameter values, but here for the double-site out-of-phase compacton solution; also the bottom right panel here corresponds to $t=180$. The instability strength for this solution is $max\left[\mathrm{real}\right(\nu \left)\right]\approx 0.0298$. Notice that here the eventual evolution leads to a single-site excitation (see the amplitude evolution in the top left panel and the eventual profile in the bottom right panel).

Figure 9

This figure is similar to Fig. 4, but here the parameter values common across all four plots are $L=11,\epsilon =0.01$, and $\mathrm{\Omega }={\gamma }_1=10$ so that $\alpha =1$. The leftmost plot is associated with a four-site symmetric solution using the first zero of the Bessel function with $j=1$, the middle plot corresponds to a four-site antisymmetric solution with $j=1$, and the rightmost plot is for the five-site solution with $j=5$. The axis labels are the same as those established in Fig. 4.