Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multidimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasiperiodic, single waveguide, etc. We show that when the soliton is unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to a focusing instability, whereas violation of the spectral condition leads to a drift instability. We also present a quantitative approach that allows one to predict the stability and instability strength.

Figure 1

The spectrum of $L_{+}$ in a homogeneous medium.

Figure 2

The spectrum of $L_{+}$ in an inhomogeneous medium.

Figure 3

(Color online) The sinusoidal square lattice given by Eq. (19) with $V_0=5$. (a) Top view. (b) Side view. The solitons investigated below are centered at the lattice maximum (0,0), lattice minimum (0.25,0.25), and saddle point (0.25,0).

Figure 4

(Color online) (a) Power and (b) perturbed-zero eigenvalues, as functions of the propagation constant, for solitons centered at a maximum (blue, dashes) and minimum (red, dots) of the lattice (19) with $V_0=5$. Also shown are the corresponding lines for the homogeneous NLS equation (solid, green). The circles (black) correspond to the values used in Figs. 5, 6, 7, 8.

Figure 5

(Color online) Normalized peak intensity (18) of solutions of Eq. (15) with the periodic lattice (19) with $V_0=5$. Initial conditions are power-perturbed solitons [see Eq. (16)] centered at a lattice minimum: (a) Soliton from the stable branch $(\mu =-31)$. (b) Soliton from the unstable branch $(\mu =-3)$. Input powers are 0.5% (red dots), 1% (blue dashes), and 2.5% (solid green line) above the soliton power.

Figure 6

(Color online) Dynamics of solutions of Eq. (15) with the periodic lattice (19) with $V_0=5$. Initial conditions are position-shifted solitons [see Eq. (17)] centered at a lattice minimum, with $(\Delta x_0,\Delta y_0)=(0,0.04)$. (a1) Center of mass in the $y$ coordinate (blue, solid line) and analytical prediction [Eq. (20), red, dashes] for $\mu =-31$; (a2) Normalized peak intensity (18) for $\mu =-31$; (b1) and (b2) are the same as (a1) and (a2), but for $\mu =-3$.

Figure 7

(Color online) Same as Fig. 5a for a soliton at a lattice maximum with $\mu =-5$ (stable branch) and input power that is 0.5% (red dots) and 1% (blue dashes) above the soliton power.

Figure 8

(Color online) Dynamics of a soliton at a lattice maximum with $\mu =-5$, which is position-shifted according to Eq. (17) with $(\Delta x_0,\Delta y_0)\approx (0,0.02)$. (a) Center of mass in the $y$ coordinate (blue, dashes) and the analytical prediction (Eq. (21) with ${\Omega }_y\approx 3.9$ (solid black line). Location of lattice minimum and maxima are denoted by thin magenta and black horizontal lines, respectively. (b) Normalized peak intensity.cos

Figure 9

(Color online) (a) Same as Fig. 4b for solitons centered at a saddle point. One eigenvalue is shifted to negative values (black, solid), and is indistinguishable from the eigenvalue at lattice minima (blue, dashes); one eigenvalue is shifted to positive values (magenta, solid), and is indistinguishable from the eigenvalue at lattice maxima (red, dots); (b1) Same data as in Fig. 4a, with the addition of data for solitons centered at a saddle point of the lattice (black, dash-dots). (b2) same as (b1) showing only the data for solitons centered at a saddle point (dash-dots) and for the homogeneous medium soliton (green, solid).

Figure 10

(Color online) Dynamics of a soliton centered at a saddle point $(x_0,y_0)=(0.25,0)$ of the lattice (19) with $\mu =-12$ and shifts along: (i) the stable $x$ direction [$(\Delta x_0,\Delta y_0)\approx (0.0156,0)$, red dots], (ii) the unstable $y$ direction [$(\Delta x_0,\Delta y_0)\approx (0,0.0156)$, green dashes], and (iii) the diagonal direction [$(\Delta x_0,\Delta y_0)\approx (0.0156,0.0156)$, blue, dash-dots]. (a) Center of mass $⟨x⟩$. (b) Center of mass $⟨y⟩$. (c) Normalized peak intensity.

Figure 11

(Color online) The shallow maximum periodic lattice given by Eq. (23) with $V_0=5$. (a) Top view. (b) Side view. (c) Cross section along the line $x=y$.

Figure 12

(Color online) Same as Fig. 4 for solitons centered at a shallow local maximum of the shallow-maximum periodic lattice (23).

Figure 13

(Color online) Dynamics of a perturbed soliton at shallow-maximum periodic lattice (23) with a narrow soliton [(a1) and (a2) with $\mu =-12$] and a wide soliton [(b) with $\mu =-2$], and using $(\Delta x_0,\Delta y_0)=(0.05,0.05)$. (a1) Center of mass $⟨x⟩=⟨y⟩$ of the narrow soliton (blue, dashes) and the analytical prediction (red dots). (a2) Normalized peak intensity of the narrow soliton. (b) Same as (a1) for the wide soliton.

Figure 14

(Color online) Same as Fig. 11 for solitons centered at the “vacancy” of the lattice (24).

Figure 15

(Color online) Same as Fig. 4 for solitons at the vacancy of the lattice (24). (b) The perturbed-zero eigenvalues ${\lambda }_0^{\left(1\right),\left(2\right)}$ are slightly different from each other. The circles (black) correspond to the values used in Fig. 16.

Figure 16

(Color online) Same as Fig. 13 but for the vacancy lattice (24). Here ${\Omega }_x\approx 3$ in (a2) and ${\Omega }_x\approx 1.09i$ in (b). In both cases the $⟨y⟩$ dynamics (not shown) is similar (but not identical) to the $⟨x⟩$ dynamics.

Figure 17

(Color online) (a)–(c) Contours of the intensity ${\mid u(x,y,z)\mid }^2$ (blue) superimposed on the vacancy lattice (red) with initial conditions corresponding to the mode with $\mu =-8$ that is initially shifted in the $(x,y)$ plane to $(\Delta x_0,\Delta y_0)=(0.05,0.1)$, i.e., at an angle of 63° to the $y$ axis. (a) $z=0$, $I\approx 1$, (b) $z=0.51$, $I\approx 2.18$, and (c) $z=0.63$, $I\approx 11.1$. (d) Center of mass dynamics (black curve) and the analytical prediction (blue, dashes) superimposed on the contours of the potential (red). (e) $⟨x⟩$ (blue, solid) and $⟨y⟩$ (red, dashes) as functions of $I\left(z\right)$. Circles (black) correspond to the $z$-slices shown in (a)–(c).

Figure 18

(Color online) Same as Figs. 11a, 11b for the Penrose quasicrystal lattice given by Eq. (26) with $N=5$ and $V_0=5$.

Figure 19

(Color online) Same as Fig. 4 for solitons at the maxima of the lattices (26) with $N=4$ (periodic lattice, dashed blue line), $N=5$ (Penrose quasicrystal lattice, dash-dotted red line), $N=11$ (higher-order quasicrystal lattice, dotted black line), and the homogeneous NLS soliton (solid green line).