Frustrated Brownian Motion of Nonlocal Solitary Waves

We investigate the evolution of solitary waves in a nonlocal medium in the presence of disorder. By using a perturbational approach, we show that an increasing degree of nonlocality may largely hamper the Brownian motion of self-trapped wave packets. The result is valid for any kind of nonlocality and in the presence of nonparaxial effects. Analytical predictions are compared with numerical simulations based on stochastic partial differential equations.

Figure 1

Amplitude of the fluctuations $C$ for the exponential nonlocality versus ${\sigma }^2$, after Eq. (17), for various soliton powers $P$.

Figure 2

(a) Typical dynamics of a solitary wave [Eq. (3)] as obtained from the numerical solutions of (4) in the presence of randomness and exponential nonlocality ($P=6$, ${\sigma }^2=5$, ${\eta }_N=0.01$). (b) Center-of-mass trajectories of the same bound state ($P=6$, ${\sigma }^2=5$) for 100 disorder realizations (${\eta }_N=0.01$).

Figure 3

Comparison of the numerically (continuous lines, $P=6$, ${\eta }_N=0.01$) and theoretically (circles for ${\sigma }^2=2$, squares for ${\sigma }^2=5$, and diamonds for ${\sigma }^2=10$) calculated standard deviation of the solitary-wave position versus the evolution coordinate for three degrees of nonlocality and 100 disorder realizations.