Soliton trapping in a disordered lattice

In recent years, the competition between randomness and nonlinearity was extensively explored. In the present paper, the dynamics of solitons of the Ablowitz-Ladik model in the presence of a random potential is studied. In the absence of the random potential, it is an integrable model and the solitons are stable. As a result of the random potential, this stability is destroyed. In a certain regime, for short times, particlelike dynamics with constant mass is found; in another regime, particlelike dynamics with varying mass takes place. In particular, an effective potential is found that predicts correctly changes in the direction of motion of the soliton. This potential is a scaling function of time and strength of the potential, leading to a relation between the first time when the soliton changes direction and the strength of the random potential.

Figure 1

AL soliton acceleration by one specific realization of the random potential [${\mu }_0=3, v(t=0)=1$, and $W=0.1$]. (a) Comparison of the center of mass between the semianalytical (solid green line) and numerical (dashed red line) results. (b) Comparison of the semianalytical (lower solid green line) and numerical (lower dashed red line) velocities, and the second moment $m_2$ (upper solid blue line). (c)–(e) Zoomed views of the velocity comparison in three different time intervals. (f) The soliton profiles at $t=0$ (dashed blue line) and $t=1000$ (solid red line). Note that the site coordinate is fixed on the center of mass.

Figure 2

AL soliton trapping by one realization of the random potential [${\mu }_0=1, v(t=0)=1$, and $W=0.1$]. (a) Center of mass $x$ [Eq. (12)] as a function of time $t$. (b) The soliton velocity $v$ [Eq. (14)] as a function of time $t$. (c) The mass $M_s$ [Eq. (11)] as a function of time $t$. (d) The soliton profiles at $t=0$ (dashed blue line) and $t=3000$ (solid red line). (e) $v^{\left(\text{av}\right)}$ and $M_s^{\left(\text{av}\right)}$ as a function of time $t$ (the averaging is performed over 12 realizations).

Figure 3

AL soliton propagation in one realization of the random potential [${\mu }_0=0.5, v(t=0)=1$, and $W=0.1$]. (a) The soliton profiles at $t=0$ (dashed blue line) and $t=1000$ (solid red line). (b) The mass $M_s$ [Eq. (11)] as a function of time $t$. (c) The soliton velocity $v$ [Eq. (14)] as a function of time $t$. (d) $v^{\left(\text{av}\right)}$ and $M_s^{\left(\text{av}\right)}$ as a function of time $t$ (the averaging is performed over 12 realizations).

Figure 4

(a) Comparison of the two forces $F_1$ [Eq. (21)] (solid blue line) and $F_2$ [Eq. (22)] (dashed red line) with the soliton parameters and random potential the same as those in Figs. 2 and 2. (b)–(d) Zoomed views of panel (a) for three different time intervals. (e) The effective potential $U$ (solid red line) and soliton velocity $v$ (dotted green line) after the soliton's first reflection ($T_c\approx 1100$).

Figure 5

(a) $T_c$ as function of $W$. The random potentials are constructed of one realization of random numbers in $[-1,1]$, multiplied by different strength $W/2$. (b) Linear fit of the data (logarithm forms of variables) in (a). (c) $U^{\left(\text{av}\right)}$ as a function of $t$ and $W$ (the lines from left to right correspond to $W$ from 0.10 to 0.06). (d) $U^{\left(\text{av}\right)}$ as a function of the scaling variable $t{W}^2$.

Figure 6

$T_c^{\left(\text{av}\right)}$ (blue circles), $x^{\left(\text{av}\right)}$ (green squares), and ${\sigma }_x$ (red stars) as functions of $W$ (linear fit of the logarithm of variables). The averaging is carried out over 250 realizations of the random potential. The initial soliton parameters are ${\mu }_0=1$ and $v(t=0)=1$.

Figure 7

Comparison of the typical results, as for parameters of Figs. 2 and 3, integrating on Eq. (8) by the fourth-order Runge-Kutta method with different step sizes [$\mathrm{\Delta }t={10}^{-3}$ (solid blue line) and $\mathrm{\Delta }t={10}^{-4}$ (dotted green line)] and the Besse method with $\mathrm{\Delta }t={10}^{-3}$ (dashed red line). (a,b) ${\mu }_0=1, v(t=0)=1$, and $W=0.1$ used in Fig. 2 with $\mathrm{\Delta }t={10}^{-3}$; (c,d) ${\mu }_0=0.5, v(t=0)=1$, and $W=0.1$ used in Fig. 3 with $\mathrm{\Delta }t={10}^{-3}$. The RK results of $\mathrm{\Delta }t={10}^{-2}$ look similar.