Mobility of Discrete Solitons in Quadratically Nonlinear Media

We study the mobility of solitons in lattices with quadratic (${\chi }^{(2)}$, alias second-harmonic-generating) nonlinearity. Using the notion of the Peierls-Nabarro potential and systematic numerical simulations, we demonstrate that, in contrast with their cubic (${\chi }^{(3)}$) counterparts, the discrete quadratic solitons are mobile not only in the one-dimensional (1D) setting, but also in two dimensions (2D), in any direction. We identify parametric regions where an initial kick applied to a soliton leads to three possible outcomes: staying put, persistent motion, or destruction. On the 2D lattice, the solitons survive the largest kick and attain the largest speed along the diagonal direction.

Figure 1

The left panel shows the dependence on the coupling strength of the free-energy difference $\Delta F$ between intersite and on-site positions of the 2D discrete soliton (shifted in one lattice direction), for $\omega =-0.25$ and $k=0.25$. The dashed line is the best fit, for large $C$, to the analytical prediction, $aC\exp (-bC)$, see text. The right panel shows the comparison of $\Delta F$ and $2\Delta F$ for the soliton set intersite, respectively, in both directions (solid line), and in one direction only (dashed line).

Figure 2

Space-time contour plots of $|{\psi }_{m,n}{|}^2$ and $|{\varphi }_{m,n}{|}^2$ for the FF and SH fields (top and bottom panels) in the 1D lattice with periodic boundary conditions, for $C_1=C_2=1$, $\omega =-0.25$, $k=0.25$. The kick, $S=0.4$ or 3.0 (left and right parts), sets the soliton in stable motion, or destroys it, respectively.

Figure 3

A diagram in the plane of the coupling strength, $C=C_1=C_2$, and kick size, $S$, showing outcomes of kicking the quiescent soliton in the 1D (left panel) and 2D (right panel) lattice, for $\omega =-0.25$ and $k=0.25$. Solid lines show exponential fitting curves, $S=6.02389\sqrt{C_2}\exp (-13.7684C_2)$ and $S=2.56619\sqrt{C_2}\exp (-9.0043C_2)$, for the 1D and 2D cases, respectively. In the 2D case, triangles and rectangles denote the numerical results for $\theta =20°$ and $\theta =45°$, respectively. (localization means immobility).

Figure 4

Motion of the discrete soliton in a 2D periodic lattice (same parameters as in Fig. 2). Depicted are the cases of propagation along the diagonal (45°, top panels) and at an arbitrarily chosen direction (20°, bottom panels). In the left panels, dashed and solid lines show coordinates of the soliton’s center, $\xi$ and $\eta$, as functions of time (in the case of the diagonal motion, $\xi =\eta$). In the right panels, snapshots of the moving soliton in the FF (top) and SH (bottom) fields at $t=0$, 30, 50 are shown.

Figure 5

Characteristics of the soliton motion in the 2D periodic lattice, for $C_1=C_2=1$, $k=0.25$, $\omega =-0.25$. The top panel shows the velocity versus strength $S$ of the kick directed along the lattice bonds (at angle $\theta =0$); the vertical dashed line indicates the value $S_{\mathrm{cr}}$, beyond which the soliton is destroyed. The bottom left panel depicts $S_{\mathrm{cr}}$ as a function of orientation $\theta$ of the initial kick. For given $S=0.4$, the velocity is shown vs $\theta$ in the bottom right corner.