Solitonization of the Anderson localization

We study the affinities between the shape of the bright soliton of the one-dimensional nonlinear Schroedinger equation and the shape of the disorder-induced localization with a Gaussian random potential. For the most localized state, we derive expressions for the nonlinear eigenvalue and for the localization length. We investigate the stability and superlocalizations.

Figure 1

(a) For a realization of disorder, the nonlinear Anderson states $\left|\phi \right|/\text{max}\left(\right|\phi \left|\right)$ for different $P$ ($V_0=4$), two values of $P$ are indicated (blue [gray] and black dashed lines); the inset shows the projection on the soliton profile. (b) $E$ vs $P$ ($V_0=4$) for several realizations (cyan [light gray] thin lines), with the strong (red [gray] thick line) and weak (green [gray] thick line, for a single realization) expansions, and with the annealed phase-space variational approach (black thick line).

Figure 2

(a) $l$ vs $P$ ($V_0=1$): Various disorder realizations are shown (cyan [gray] thin lines) and compared with the weak (green [gray] line, for a single realization with $P_0$ indicated), with the strong expansion (red [gray] thick line), and with the variational approach (black thick line). (b) Distribution of $P_0$ (200 realizations); the red (light gray) vertical line is the analytical $P_C$. See text ($V_0=2$).

Figure 3

Stability of the nonlinear Anderson states: (a) dashed, nonlinear state at power $P=7$; full lines, two different superlocalizations arbitrarily vertically shifted (blue [gray] $\Omega =11$, green [gray] $\Omega =14$); (b) as in panel (a) for $P=23$ (blue [gray] $\Omega =59$, green [gray] $\Omega =61$); (c) eigenvalues ves power $P$ ($V_0=10$); (d) evolution of the nonlinear states with $P=3.54$ with $10\%$ amplitude noise; and (e) as in panel (d) with $P=13.2$; the white line corresponds to the arbitrarily scaled initial profile $\phi$ ($V_0=2$).