Peierls-Nabarro energy surfaces and directional mobility of discrete solitons in two-dimensional saturable nonlinear Schrödinger lattices

We address the problem of directional mobility of discrete solitons in two-dimensional rectangular lattices, in the framework of a discrete nonlinear Schrödinger model with saturable on-site nonlinearity. A numerical constrained Newton-Raphson method is used to calculate two-dimensional Peierls-Nabarro energy surfaces, which describe a pseudopotential landscape for the slow mobility of coherent localized excitations, corresponding to continuous phase-space trajectories passing close to stationary modes. Investigating the two-parameter space of the model through independent variations of the nonlinearity constant and the power, we show how parameter regimes and directions of good mobility are connected to the existence of smooth surfaces connecting the stationary states. In particular, directions where solutions can move with minimum radiation can be predicted from flatter parts of the surfaces. For such mobile solutions, slight perturbations in the transverse direction yield additional transverse oscillations with frequencies determined by the curvature of the energy surfaces, and with amplitudes that for certain velocities may grow rapidly. We also describe how the mobility properties and surface topologies are affected by inclusion of weak lattice anisotropy.

Figure 1

Examples of spatial profiles for (a) one-site (OO), (b) two-site horizontal (EO), (c) two-site vertical (OE), (d) four-site (EE), (e) IS1, (f) IS2, (g) IS3-vertical, and (h) diagonal solutions, respectively, illustrated for an isotropic lattice ($\alpha =1$).

Figure 2

Properties of fundamental stationary solutions for $\gamma =4$ and $\alpha =1$: (a) Power vs frequency; (a)-inset: $\Delta P/P$ vs $\lambda$; (b) $\Delta H$ vs power; (c) stability vs power. Different solutions are plotted with different line types: one-site (thick solid blue), two-site (thick dotted blue), four-site (thick dashed blue), IS1 (thin solid black), IS2 (thin dotted black), and IS3 (thin dashed black), respectively.

Figure 3

Array scheme showing the constraint locations used to obtain the energy surfaces below.

Figure 4

Energy surfaces for $\gamma =4$, $\alpha =1$, in the five different power regimes discussed in the text : (a) $P=5$; (b) $P=9.45$; (c) $P=10$; (d) $P=12$; (e) $P=35.5$. The center of mass $\{X,Y\}$ for the four stationary solutions are $\{8,8\}$ (OO), $\{8.5,8\}$ (EO), $\{8,8.5\}$ (OE), and $\{8.5,8.5\}$ (EE). White dots denote local extrema corresponding to intermediate solutions. (System size $N=M=15$ and fixed boundary conditions were used.)

Figure 5

Examples of mobility dynamics in the propagation direction $z$, corresponding to surfaces shown in Figs. 4a, 4c, 4d, and 4e, respectively. In each subfigure, the top figures show profiles movement [(a) shows a colormap where colors were normalized to the maximum amplitude of each plot; this colormap also applies to (b)–(d)], and bottom figures show the center-of-mass evolution for $X$ (full line) and $Y$ (dashed line). (a) $P=5$, $k_x=k_y=0.038$; (b) $P=10$, $-k_x=k_y=0.018$; (c) $P=12$, $k_x=k_y=0.015$; (d) $P=35.5$, $k_x=0.015$, and $k_y=0$. (Other parameter values are the same as in Fig. 4.)

Figure 6

Dynamics of an OO mode with $P=44$ ($\gamma =4,\alpha =1$), kicked with $k_y=0.001$ and varying $k_x$. (a) Extended energy surface. Darker (brighter) regions correspond to lower (higher) values of the energy $H$. (b) $Y_9,Y_{10},Y_{11},Y_{12}$ vs $k_x$ represented by (black) filled circles, (gray) squares, (blue) diamonds, and (red) triangles, respectively. (c) $Y(z)$ vs $X(z)$ for $k_x=0.0111,0.0141,0.021,0.049$, and $0.083$ represented by thick (blue) solid, thin (black) dashed, (blue) dotted, thin (black) solid, and thick (blue) dashed lines, respectively. (d) Dashed (blue) and solid (black) lines correspond to $k_x=0.018$ and $0.035$, respectively, for $N=M=41$.

Figure 7

Density plots of $g$ vs $P$ and $\alpha$ for $\gamma =4$, for the (a) OO, (b) EO, (c) OE, and (d) EE solutions, respectively. Black means $g=0$ and lighter colors imply an increasing instability.

Figure 8

Energy surfaces for anisotropic lattices with $\gamma =4$: (a) $\alpha =0.72$, $P=19.5$; (b) $\alpha =0.7$, $P=20$.