Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity

We address the issue of mobility of localized modes in two-dimensional nonlinear Schrödinger lattices with saturable nonlinearity. This describes, e.g., discrete spatial solitons in a tight-binding approximation of two-dimensional optical waveguide arrays made from photorefractive crystals. We discuss the numerically obtained exact stationary solutions and their stability, focusing on three different solution families with peaks at one, two, and four neighboring sites, respectively. When varying the power, there is a repeated exchange of stability between these three solutions, with symmetry-broken families of connecting intermediate stationary solutions appearing at the bifurcation points. When the nonlinearity parameter is not too large, we observe good mobility and a well-defined Peierls-Nabarro barrier measuring the minimum energy necessary for rendering a stable stationary solution mobile.

Figure 1

(Color online) (a) OO profile. (b) OE profile. (c) EE profile. (d) and (e) $P$ versus $\lambda$ for $\gamma =4$ and for $\gamma =10$, respectively. Diamonds, stars, and squares represent OO, OE, and EE solutions, respectively. System size $N=M=15$ [(d) and (e) insets: $N=M=31$].

Figure 2

(Color online) Results for $\gamma =4$. (a) Smallest eigenvalue $G$ versus power. (b) Close-up of (a) in the region $P\sim 9.4$. (c) Profile of intermediate solution (IS). Diamonds, stars, squares, and triangles represent OO, OE, EE, and IS solutions, respectively.

Figure 3

(Color online) Results for $\gamma =10$. (a) Smallest eigenvalue $G$ versus power. (b) Profile of intermediate solution (IS). (c) Energy $H$ versus power for the region enclosed with a circle in (a). Diamonds, stars, squares, and triangles represent OO, OE, EE, and IS solutions, respectively.

Figure 4

(Color online) Mobility results. $\gamma =4$, $P=3.99$: (a) axial propagation for $k_x$ or $k_y=0.0323$, (b) OO profile, and (c) diagonal propagation for $k_x=k_y=0.031$. $\gamma =4$, $P=9.42$: (d) axial propagation for $k_x$ or $k_y=0.00624$ and (e) OO profile. $\gamma =10$: (f) axial propagation for $k_x$ or $k_y=0.6$ $(P=20.43)$, (g) axial propagation for $k_x$ or $k_y=0.5$ $(P=24.76)$, and (h) OO profile. System size $N=M=15$, periodic boundary conditions.