Stability of solitary waves in random nonlocal nonlinear media

We consider the interplay between nonlocal nonlinearity and randomness for two different nonlinear Schrödinger models. We show by means of both numerical simulations and analytical estimates that the stability of bright solitons in the presence of random perturbations increases dramatically with the nonlocality-induced finite correlation length of the noise in the transverse plane. In fact, solitons are practically insensitive to noise when the correlation length of the noise becomes comparable to the extent of the wave packet. We characterize soliton stability using two different criteria based on the evolution of the Hamiltonian of the soliton and its power. The first criterion allows us to estimate a time (or distance) over which the soliton preserves its form. The second criterion gives the lifetime of the solitary wave packet in terms of its radiative power losses. We derive a simplified mean field approach which allows us to calculate the power loss analytically in the physically relevant case of weakly correlated noise, which in turn serves as a lower estimate of the lifetime for correlated noise in the general case.

Figure 1

Single realizations of the propagation of local solitons perturbed by correlated noise for different values of $\sigma$ (degree of nonlocality). In (a)–(d), we have $\sigma =3,$ 1, 0.5, $1/128$, with the respective coupling strength $n=\sqrt{\sigma /5}$. The decay of the soliton due to the emission of radiation becomes stronger for smaller $\sigma$, even though the coupling strength is reduced as well. Apart from the decay, solitons perform a random walk in the transverse plane.

Figure 2

(a) Evolution of ${\left|\langle \psi (x=0,t)\rangle \right|}^2$ (solid lines) compared to the analytical prediction Eq. (11) (dashed line). The correlation length is $\sigma =\infty$, 3, 1, and 1/128 as indicated in the plot for the numerical solid lines. The autocorrelation $C\left(0\right)=n^2/4\sigma =1/20$ is kept constant. (b) The corresponding soliton random walk (solid lines) compared to the analytical prediction of Eq. (14) (dashed lines). In our simulations ensemble averages are taken over 128 realizations.

Figure 3

(a) Evolution of Hamiltonian $\mathcal{H}$ [Eq. (15)] (solid lines) compared to analytical predictions from Eq. (16) (dashed lines). The correlation length is $\sigma =3$, 1, and 0.5 as indicated in the plot. The autocorrelation $C\left(0\right)=n^2/4\sigma =1/20$ is kept constant. (b) The analytical prediction for $t_{\mathcal{H}}$ (dashed black line) as a function of $\sigma$ is shown. Numerical values are indicated by circles, where the colored circles correspond to the colors used in (a). In our simulations, ensemble averages are taken over 128 realizations.

Figure 4

(a) Time $t_{1/e}$ where the soliton power drops below $P_{\mathrm{S}}(t=0)/e$ versus degree of nonlocality $\sigma /{\sigma }_{\mathrm{S}}$. Numerical simulations confirm the analytical prediction Eq. (19), $t_{1/e}=1/C\left(0\right)$, in the weakly nonlocal limit $\sigma /{\sigma }_{\mathrm{S}}\ll 1$. The dotted line represents a guide to the eye $\propto {(\sigma /{\sigma }_{\mathrm{S}})}^{3/2}$. The autocorrelation $C\left(0\right)=n^2/4\sigma =1/20$ is kept constant. (b) The corresponding evolution of the power of the solitons is shown. The correlation lengths used are $\sigma =$ 3, 1, 0.5, and 1/128 for the upper violet, then black, green and finally the lowest red curve, respectively. In our simulations, ensemble averages are taken over 128 realizations.

Figure 5

(a) Evolution of the ensemble-averaged peak intensity $\langle |\psi {|}_{\max }^2\rangle$ (solid red lines) compared to analytical predictions Eq. (19) (dashed black lines) for $\delta$-correlated white noise. The coupling strength is $n=0.005$, 0.01, 0.02, 0.03, and $\sqrt{0.05\Delta x}$ from the top to the lowest curve, respectively; the spatial step size $\Delta x=80/2048$ is kept constant. The last value $n=\sqrt{0.05\Delta x}$ is chosen such that it yields the same decay rate $C\left(0\right)$ for the soliton power as the case $\sigma =1/128$ in Fig. 4. (b) The corresponding evolution of the power of the solitons is shown. In our simulations, ensemble averages are taken over 128 realizations.

Figure 6

(a) Dependence of the $1/e$ width ${\sigma }_{\mathrm{S}}$ of the nonlocal soliton as a function of power $P_{\mathrm{S}}$ and degree of nonlocality $\sigma$. (b) Exemplary intensity profiles of nonlocal solitons for different values of $\sigma$ and fixed power $P_{\mathrm{S}}=4$.

Figure 7

Soliton dynamics in nonlinear nonlocal model with additive noise Eq. (1). (a) Numerically found evolution of the averaged Hamiltonian $\langle \mathcal{H}\rangle$ [Eq. (24)] (solid lines) compared with analytical predictions from Eq. (16) (dashed lines). The nonlocal length is $\sigma =3$, 1, 0.5, and $1/128$ as indicated. The autocorrelation $C\left(0\right)=n^2/4\sigma =1/20$ is kept constant. (b) Corresponding evolution of the power of the solitons. (c) Analytically predicted $t_{\mathcal{H}}$ (dashed black line) as a function of $\sigma$. Numerical values are indicated by circles, where the colored circles correspond to the colors used in (a) and (b). The inset in (c) shows the functional dependency of the relative correlation length $\sigma /{\sigma }_{\mathrm{S}}$ on the nonlocal length $\sigma$ for $P_{\mathrm{S}}=4$. (d) Soliton lifetime $t_{1/e}$ where the soliton power drops below $P_{\mathrm{S}}(t=0)/e$ versus degree of nonlocality $\sigma /{\sigma }_{\mathrm{S}}$. Numerical simulations confirm the analytical prediction Eq. (25), $t_{1/e}=1/C\left(0\right)$, in the weakly nonlocal limit $\sigma /{\sigma }_{\mathrm{S}}\ll 1$. The dotted line represents a guide to the eye $\propto {(\sigma /{\sigma }_{\mathrm{S}})}^2$. In our simulations, ensemble averages are taken over 128 realizations.

Figure 8

Nonlinear nonlocal model with multiplicative noise Eq. (26). (a) Soliton random walk (solid lines) compared to the analytical prediction of Eq. (31) (dashed lines). (b) The corresponding evolution of the power of the solitons is shown. The correlation length is $\sigma =\infty$, 3, 1, and 1/128 as indicated. The coupling strength $n$ is adjusted such that $n^2/4\sigma =1/20$ is kept constant. (c) Numerical values of $t_{\mathcal{H}}$ as a function of $\sigma$ are shown. Colored circles correspond to the colors used in (a) and (b). The inset in (c) shows the functional dependency of the relative correlation length $\sigma /{\sigma }_{\mathrm{S}}$ on the nonlocal length $\sigma$ for $P_{\mathrm{S}}=4$. (d) Numerical values of the time $t_{75\%}$ where the soliton power drops below $0.75P_{\mathrm{S}}(t=0)$ versus degree of nonlocality $\sigma /{\sigma }_{\mathrm{S}}$ are shown. In our simulations, ensemble averages are taken over 128 realizations.