Abstract

We numerically investigate the transport of discrete breather-like states (DBSs) in a nonlinear optical lattice with weak disorder. We find that the DBS’s center of mass experiences a transient anomalous diffusion before its localization. This diffusive process is shown to represent the property of weak ergodicity breaking.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Discrete breathers (DBs), the spatially localized and time-periodic stable excitations, emerge as a prototype class of nonlinear waves in lattice systems [1,2]. They were first observed experimentally through light propagation in 1D waveguide arrays [3], and are currently studied in many areas such as Bose-Einstein condensates (BECs) [4] and crystal physics [5]. The mobile DBs can transport energy in a lattice, where they may need to overcome the so-called Peierls-Nabarro barriers [6]. Moreover, interaction of the moving DBs with ordered lattice impurity leads to various phenomena, including reflection, transmission, or trapping of these DBs [7].

Disorder, on the other hand, can suppress the diffusive transport of wave packets, known as the famous Anderson localization [810]. In 2007, the first experiment of Anderson localization was successfully performed by Schwartz et al. and Lahini et al. in disordered photonic lattices [11,12], where they also show the localization can be enhanced under self-focusing nonlinearity. Nevertheless, numerical simulations in 1D tight-binding systems reveal that, Anderson localization can be destroyed by weak nonlinearity, giving rise to a subdiffusive spreading of the wave packets [13,14]. For large nonlinearities, self-trapping of a fraction of the wave packets occurs [1416], which was related to the emergence of breather-like excitations.

In this paper, facing an aspect of the interaction between disorder and strong nonlinearity, we study the transport of discrete breather-like states (DBSs) in a disordered optical lattice. These localized states oscillate with time that may not be strictly periodic due to the presence of disorder. Related work can be traced back to 1986, when the celebrated Gordon-Haus effect was established for amplified solitons in optical fiber transmission [17]. The Brownian soliton motion (normal diffusion of solitons) was predicted in light-induced random photonic lattices [18], and was also observed recently in BECs with impurities [19]. Besides, the so-called Anderson localization of solitons were reported in random media as well [2022]. Here, we will numerically show that, the DBSs, analogous to classical particles, experience a transient anomalous diffusion with the property of non-ergodicity, in a 1D disordered nonlinear optical lattice.

2. Model and method

The evolution of optical waves in a 1D lattice of coupled waveguides along the propagation coordinate $z$ is described by the following discrete nonlinear Schrödinger (NLS) equation in its dimensionless form

$$i \frac{\partial\psi_n}{\partial z} ={-}(\psi_{n-1}+\psi_{n+1})-\nu|\psi_n|^{2}\psi_n+\epsilon_n\psi_n~,$$
where $\psi _n$ is a complex function at site $n$, and $\nu$ characterizes the strength of the cubic nonlinearity. In this work, we take $\nu =1$ (focusing nonlinearity), and consider the disorder $\epsilon _n$ normally distributed with zero mean and a standard deviation $\sigma$, i.e., $\langle \epsilon _n\rangle =0$ and $\langle \epsilon _n\epsilon _{n'}\rangle =\sigma ^{2}\delta (n-n')$, where the angular brackets stand for statistical averaging. The total power of the lattice is conserved as $P=\sum _n|\psi _n|^{2}$, and the long-lived DBs can be generated from an initially localized wave packet for the case without disorder [4,23]. Hereafter, we will employ the hyperbolic-profile excitations $\psi _n(z=0)=\sinh (\mu )/\cosh (\mu n)$ to produce the DBSs in the disordered lattice. We stress that the relatively weak disorder $\sigma =0.002$, compared with the strength of nonlinearity, is used in order to keep the DBSs neither breakup, nor radiate quickly for a considerably long time [24]. On the other hand, the parameter $\mu$ should satisfy $\mu \lesssim 0.75$ so that the DBS behaviors are dominated by the effect of disorder [25]. For most of our simulation results, we choose $\mu =0.50$ ($P=1.09$) without loss of generality.

Equation (1) was integrated by using a fourth-order Runge-Kutta scheme with periodic boundary conditions. The lattice size where $n\in [-N,N]$ was large enough with $N=500$ to make sure that the DBSs are far away from the boundaries. The discretization of the propagation distance was $\Delta z=0.01$, and the total distance was computed up to $L=10^{5}$. Our statistical results as below were obtained by averaging over $N_r=1024$ realizations, and the DBSs remained localized and robust at all times for each realization. To double check our numerical results, we varied $\Delta z$ by orders of magnitude, even changed the integration scheme by a second-order Besse method [26], and found the DBS behaviors kept the same.

3. Results and analysis

Figure 1(a) shows a typical evolution of the DBS for a specific realization of disorder, where the DBS is scattered by the disorder to experience transverse displacements, and may change its moving direction at certain spots of strong impurities. In order to study their particle-like motions, we consider random walks of the DBS’s center of mass $x(z)$ with respect to $z$, defined as $x(z)=\sum _n n|\psi _n|^{2}/P$. Here $x(z)$ physically represents the dependence of the DBS center coordinate on the propagation distance. In Fig. 1(b), $x(z)$ is plotted as functions of $z$ for $2^{10}=1024$ independent realizations of disorder, which appears to display the features of diffusion.

We use two basic quantities to characterize these diffusive motions [27]: one is the ensemble-averaged mean squared displacement (eMSD), and the other one is the time-averaged mean squared displacement (tMSD), where the coordinate $z$ plays a role of the time $t$ in our context. The eMSD is defined as

$$\langle x^{2}(z) \rangle \triangleq \frac{1}{N_r}\sum_{i=1}^{N_r} x_i^{2}(z)~.$$
For classical particles, the eMSD can display either a Brownian motion $\langle x^{2}(z) \rangle \thicksim z$, or an anomalous diffusion $\langle x^{2}(z) \rangle \thicksim z^{\alpha }$ with $\alpha \neq 1$ for a certain scale of $z$. Anomalous diffusion is an important subject in complex systems, and has been studied in various occasions [2729]. In Fig. 2(a), we plot the eMSD $\langle x^{2}(z)\rangle$ as a function of $z$ on a log-log scale for $5$ orders of magnitude, with the same representative parameters in Fig. 1. The eMSD seems to increase monotonically with $z$ exceeding $z=3000$, corresponding to a diffusive process, and then reaches a plateau (localization of DBSs) for the longer $z$. The diffusion covers several intervals with different scaling exponents $\alpha$ for each interval, as seen in Table 1. We can apparently find a transient anomalous diffusion, that includes an initial stage with $\alpha \approx 3.3$, an intermediate stage with $\alpha \approx 1.5$, and a final stage with $\alpha \approx 0.8$.

 figure: Fig. 1.

Fig. 1. (a) Evolution of the DBS in the lattice for a specific realization of disorder; (b) Center of mass of the DBSs $x(z)$ for 1024 independent realizations of disorder. The parameters are $\mu =0.50$, $\sigma =0.002$, and $\nu =1$.

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 figure: Fig. 2.

Fig. 2. (a) The eMSD $\langle x^{2}(z)\rangle$ as a function of $z$; (b) The tMSDs $\overline {\delta ^{2}(\Delta )}$ (red thin curves) for individual trajectories as functions of $\Delta$, and the trajectory-averaged tMSD $\left \langle \overline {\delta ^{2}(\Delta )} \right \rangle$ is denoted by the black bold curve. All these quantities are evaluated from $N_r=1024$ independent realizations. The parameters are the same as in Fig. 1 ($L=10^{5}$).

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Tables Icon

Table 1. Diffusive property of the DBS’s center of mass. For different intervals of $z$, $\alpha$ and $\alpha '$ are obtained by respectively fitting the eMSD and tMSD curves to $\langle x^{2}(z)\rangle \thicksim z^{\alpha }$ and $\left \langle \overline {\delta ^{2}(\Delta )}\right \rangle \thicksim \Delta ^{\alpha '}$. The $R^{2}>0.997$ is kept for every fit.

On the other hand, from the viewpoint of single particle tracking, the tMSD of an individual trajectory is given by [27]

$$\overline{\delta^{2}(\Delta,L)} \triangleq \frac{1}{L-\Delta} \int_0^{L-\Delta} [x(z+\Delta)-x(z)]^{2}\textrm{d}z~,$$
with a variable $\Delta \ll L$ defining a window slid along the data series $x(z)$ (hereafter we use $0<\Delta \leqslant L/10$). In Fig. 2(b), the log-log plot of tMSDs $\overline {\delta ^{2}(\Delta )}$ as functions of $\Delta$ for $1024$ trajectories is presented, where we see the tMSDs vary from trajectory to trajectory, and show an obvious scatter. This gives the fact that the time averages remain random even for long averaging time, and an individual time-averaged physical observable cannot be reproduced from another one. We therefore compute the trajectory-averaged tMSD $\left \langle \overline {\delta ^{2}(\Delta )}\right \rangle$, as fitted in Table 1, which displays the feature of a transient anomalous diffusion as well.

However, comparison of the eMSD and tMSDs clearly reveals the inequality

$$\left\langle\overline{\delta^{2}(\Delta)}\right\rangle \neq \langle x^{2}(\Delta)\rangle~,$$
for different intervals of the diffusive process ($\Delta \ll L$), with the identification $z\leftrightarrow \Delta$. For the particle-like motion, this inequivalence indicates occurrence of weak ergodicity breaking, meaning that the information obtained in the time average is different from that given by the ensemble average [27,30,31]. This should be important since it reminds us to be careful when we draw some conclusions from a single-trajectory measurement of DBSs with long duration. For example, regarding the interval $z\in [400,1000]$, the tMSD gives rise to the seemingly subdiffusive behavior with $\alpha '\approx 0.9$, as opposed to the eMSD-suggested superdiffusive transport with $\alpha \approx 1.5$. In principle, such a statement of non-ergodicity should be made in the limit of $L\rightarrow \infty$, however, the computational domain $L=10^{5}$ is far larger than the diffusive domain of $z\lesssim 3000$, and the disparity between the eMSD and the tMSDs seems to be kept for longer $L$.

To further understand the scatter of the tMSDs $\overline {\delta ^{2}(\Delta )}$ between individual trajectories, we can investigate the distributions of these tMSDs by using the probability density function (PDF) $\phi (\xi )$ of the following normalized ratio

$$\xi \triangleq \frac{\overline{\delta^{2}(\Delta,L)}}{\left\langle\overline{\delta^{2}(\Delta,L)}\right\rangle}~.$$
For an ergodic process we have the PDF $\phi (\xi )\rightarrow \delta (\xi -1)$ for long measured trajectories, which indicates that individual time-averaged trajectories can be reproducible. Oppositely, deviation from this type of symmetric form may imply non-ergodic processes even for relatively short trajectories [32]. In Fig. 3 we show the distribution histograms for $\xi$ estimated from different diffusion intervals. The asymmetric shape of the distribution $\phi (\xi )$ (around $\xi =1$) can be observed basically, and the distribution becomes wider for longer $\Delta$, which appear to be the characteristic feature of ergodicity breaking [27]. On the other hand, with $\Delta$ dramatically increasing, the extreme events ($\xi$ far deviating from the unity, i.e., $\xi \gg 1$) increase apparently, and the value of $\phi (\xi )$ near $\xi =0$ enhances as well. These facts mean that a higher percentage of the DBSs cannot diffuse effectively at longer propagation distance, and the extreme events contribute to the trajectory-averaged observables remarkably. Consequently, beside the violation of ergodicity, Fig. 3 also implies that binding of the DBSs by disorder would be a primary mechanism for the transition between different diffusive intervals. An intuitive observation of such a mechanism may be found from massive trajectories of the DBSs as well [see Fig. 1(b) for instance].

 figure: Fig. 3.

Fig. 3. Scatter distribution $\phi (\xi )$ for the DBS’s diffusive process. Panels (a)-(c) are extracted at $\Delta =20$, $\Delta =450$, and $\Delta =1250$, corresponding to the diffusion intervals with different exponents $\alpha$. The parameters are the same as in Fig. 2.

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For longer propagation distance ($z\gtrsim 5000$), the DBSs become trapped, displaying the transverse Anderson-like localization of DBSs. In general, increasing the strength of the disorder may shorten the transient diffusive process, resulting the DBS motions confined in a narrower spatial region.

Since the nonlinear localized states, such as solitons, may be used as the information carriers in optical communication systems, the studies on how their propagations are influenced by noise and random inhomogeneities become necessary. One example is the soliton Gordon-Haus jitter for which the distributed noise leads the variance of soliton displacement to satisfy $\langle x^{2}\rangle \thicksim z^{3}$ [17]. In this work our simulation of the DBS propagation in 1D disordered waveguide lattices reveals the existence of intervals where the displacement variance grows in a slower way, $\langle x^{2}\rangle \thicksim z^{\alpha }$ with $\alpha <3$. This suppressed transport may be realized in types of experimental setup as in [3,12]. On the other hand, we should be careful in interpreting the average of DBS trajectories since the time averages and the ensemble averages give different scaling due to ergodicity breaking for the diffusive process. The non-ergodic optical behaviors can also be found such as for Lévy flight of light in disordered media [27,33], blinking nanoscale light emitters [34], and explosions of dissipative optical solitons [35].

4. Conclusions

In summary, we numerically simulated transport of discrete breather-like states in a disordered nonlinear optical lattice, where the center of mass of the breather-like state represents a transient anomalous diffusion. Related properties were studied to suggest non-ergodicity of such the diffusive process, and binding of the DBSs by disorder was considered to be the mechanism for the diffusion transition. Our study might benefit understanding of the interaction between disorder and strong nonlinearity in optical community.

Funding

Fundamental Research Funds for the Central Universities; Beihang University (Zhuoyue Talent Program); National Natural Science Foundation of China (11902016).

Acknowledgments

The referee is appreciated for his/her valuable comments.

Disclosures

The authors declare no conflicts of interest.

References

1. D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004). [CrossRef]  

2. S. Flach and A. V. Gorbach, “Discrete breathers–advances in theory and applications,” Phys. Rep. 467(1-3), 1–116 (2008). [CrossRef]  

3. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998). [CrossRef]  

4. R. Franzosi, R. Livi, G. L. Oppo, and A. Politi, “Discrete breathers in Bose-Einstein condensates,” Nonlinearity 24(12), R89–R122 (2011). [CrossRef]  

5. S. V. Dmitriev, E. A. Korznikova, Yu A. Baimova, and M. G. Velarde, “Discrete breathers in crystals,” Phys.-Usp. 59(5), 446–461 (2016). [CrossRef]  

6. Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48(4), 3077–3081 (1993). [CrossRef]  

7. V. A. Brazhnyi, C. P. Jisha, and A. S. Rodrigues, “Interaction of discrete nonlinear Schrödinger solitons with a linear lattice impurity,” Phys. Rev. A 87(1), 013609 (2013). [CrossRef]  

8. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]  

9. M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7(3), 197–204 (2013). [CrossRef]  

10. A. Mafi, “Transverse Anderson localization of light: a tutorial,” Adv. Opt. Photonics 7(3), 459–515 (2015). [CrossRef]  

11. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007). [CrossRef]  

12. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008). [CrossRef]  

13. A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by weak nonlinearity,” Phys. Rev. Lett. 100(9), 094101 (2008). [CrossRef]  

14. S. Flach, D. O. Krimer, and C. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett. 102(2), 024101 (2009). [CrossRef]  

15. G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett. 100(8), 084103 (2008). [CrossRef]  

16. U. Naether, M. Heinrich, Y. Lahini, S. Nolte, R. A. Vicencio, M. I. Molina, and A. Szameit, “Self-trapping threshold in disordered nonlinear photonic lattices,” Opt. Lett. 38(9), 1518–1520 (2013). [CrossRef]  

17. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11(10), 665–667 (1986). [CrossRef]  

18. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Brownian soliton motion,” Phys. Rev. A 77(5), 051802 (2008). [CrossRef]  

19. L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I. Lu, V. M. Galitski, and I. B. Spielman, “Brownian motion of solitons in a Bose-Einstein condensate,” Proc. Natl. Acad. Sci. U. S. A. 114(10), 2503–2508 (2017). [CrossRef]  

20. Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E 72(2), 026606 (2005). [CrossRef]  

21. K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, “Anderson localization of solitons,” Phys. Rev. Lett. 103(21), 210402 (2009). [CrossRef]  

22. Z.-Y. Sun, S. Fishman, and A. Soffer, “Soliton trapping in a disordered lattice,” Phys. Rev. E 92(1), 012901 (2015). [CrossRef]  

23. H. Hennig, T. Neff, and R. Fleischmann, “Dynamical phase diagram of Gaussian wave packets in optical lattices,” Phys. Rev. E 93(3), 032219 (2016). [CrossRef]  

24. We address a strong nonlinearity with the ratio of the nonlinearity coefficient to the standard deviation being $500$ ($\nu /\sigma =500$). In contrast, previous studies considering the wave packet spreading by weak nonlinearity employed the ratio usually much less than $10$, even not exceeding $30$ as for the self-trapping phenomena [13–15]. The strong nonlinearity ensures that the DBSs keep their identity and highly localized no shorter than $z=10^{5}$.

25. Z.-Y. Sun, S. Fishman, and A. Soffer, “Soliton mobility in disordered lattices,” Phys. Rev. E 92(4), 040903 (2015). [CrossRef]  

26. C. Besse, “A relaxation scheme for the nonlinear Schrödinger equation,” SIAM J. Numer. Anal. 42(3), 934–952 (2004). [CrossRef]  

27. R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, “Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking,” Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014). [CrossRef]  

28. I. Golding and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. 96(9), 098102 (2006). [CrossRef]  

29. Y. Peng, L. Lai, Y.-S. Tai, K. Zhang, X. Xu, and X. Cheng, “Diffusion of ellipsoids in bacterial suspensions,” Phys. Rev. Lett. 116(6), 068303 (2016). [CrossRef]  

30. Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys. Rev. Lett. 101(5), 058101 (2008). [CrossRef]  

31. Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist,” Phys. Rev. E 81(1), 010101 (2010). [CrossRef]  

32. J.-H. Jeon and R. Metzler, “Analysis of short subdiffusive time series: scatter of the time-averaged mean-squared displacement,” J. Phys. A: Math. Theor. 43(25), 252001 (2010). [CrossRef]  

33. P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008). [CrossRef]  

34. F. Stefani, J. Hoogenboom, and E. Barkai, “Beyond quantum jumps: Blinking nanoscale light emitters,” Phys. Today 62(2), 34–39 (2009). [CrossRef]  

35. J. Cisternas, O. Descalzi, T. Albers, and G. Radons, “Anomalous diffusion of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions,” Phys. Rev. Lett. 116(20), 203901 (2016). [CrossRef]  

References

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  1. D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004).
    [Crossref]
  2. S. Flach and A. V. Gorbach, “Discrete breathers–advances in theory and applications,” Phys. Rep. 467(1-3), 1–116 (2008).
    [Crossref]
  3. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
    [Crossref]
  4. R. Franzosi, R. Livi, G. L. Oppo, and A. Politi, “Discrete breathers in Bose-Einstein condensates,” Nonlinearity 24(12), R89–R122 (2011).
    [Crossref]
  5. S. V. Dmitriev, E. A. Korznikova, Yu A. Baimova, and M. G. Velarde, “Discrete breathers in crystals,” Phys.-Usp. 59(5), 446–461 (2016).
    [Crossref]
  6. Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48(4), 3077–3081 (1993).
    [Crossref]
  7. V. A. Brazhnyi, C. P. Jisha, and A. S. Rodrigues, “Interaction of discrete nonlinear Schrödinger solitons with a linear lattice impurity,” Phys. Rev. A 87(1), 013609 (2013).
    [Crossref]
  8. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958).
    [Crossref]
  9. M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7(3), 197–204 (2013).
    [Crossref]
  10. A. Mafi, “Transverse Anderson localization of light: a tutorial,” Adv. Opt. Photonics 7(3), 459–515 (2015).
    [Crossref]
  11. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
    [Crossref]
  12. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
    [Crossref]
  13. A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by weak nonlinearity,” Phys. Rev. Lett. 100(9), 094101 (2008).
    [Crossref]
  14. S. Flach, D. O. Krimer, and C. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett. 102(2), 024101 (2009).
    [Crossref]
  15. G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett. 100(8), 084103 (2008).
    [Crossref]
  16. U. Naether, M. Heinrich, Y. Lahini, S. Nolte, R. A. Vicencio, M. I. Molina, and A. Szameit, “Self-trapping threshold in disordered nonlinear photonic lattices,” Opt. Lett. 38(9), 1518–1520 (2013).
    [Crossref]
  17. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11(10), 665–667 (1986).
    [Crossref]
  18. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Brownian soliton motion,” Phys. Rev. A 77(5), 051802 (2008).
    [Crossref]
  19. L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I. Lu, V. M. Galitski, and I. B. Spielman, “Brownian motion of solitons in a Bose-Einstein condensate,” Proc. Natl. Acad. Sci. U. S. A. 114(10), 2503–2508 (2017).
    [Crossref]
  20. Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E 72(2), 026606 (2005).
    [Crossref]
  21. K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, “Anderson localization of solitons,” Phys. Rev. Lett. 103(21), 210402 (2009).
    [Crossref]
  22. Z.-Y. Sun, S. Fishman, and A. Soffer, “Soliton trapping in a disordered lattice,” Phys. Rev. E 92(1), 012901 (2015).
    [Crossref]
  23. H. Hennig, T. Neff, and R. Fleischmann, “Dynamical phase diagram of Gaussian wave packets in optical lattices,” Phys. Rev. E 93(3), 032219 (2016).
    [Crossref]
  24. We address a strong nonlinearity with the ratio of the nonlinearity coefficient to the standard deviation being $500$500 ($\nu /\sigma =500$ν/σ=500). In contrast, previous studies considering the wave packet spreading by weak nonlinearity employed the ratio usually much less than $10$10, even not exceeding $30$30 as for the self-trapping phenomena [13–15]. The strong nonlinearity ensures that the DBSs keep their identity and highly localized no shorter than $z=10^{5}$z=105.
  25. Z.-Y. Sun, S. Fishman, and A. Soffer, “Soliton mobility in disordered lattices,” Phys. Rev. E 92(4), 040903 (2015).
    [Crossref]
  26. C. Besse, “A relaxation scheme for the nonlinear Schrödinger equation,” SIAM J. Numer. Anal. 42(3), 934–952 (2004).
    [Crossref]
  27. R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, “Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking,” Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014).
    [Crossref]
  28. I. Golding and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. 96(9), 098102 (2006).
    [Crossref]
  29. Y. Peng, L. Lai, Y.-S. Tai, K. Zhang, X. Xu, and X. Cheng, “Diffusion of ellipsoids in bacterial suspensions,” Phys. Rev. Lett. 116(6), 068303 (2016).
    [Crossref]
  30. Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys. Rev. Lett. 101(5), 058101 (2008).
    [Crossref]
  31. Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist,” Phys. Rev. E 81(1), 010101 (2010).
    [Crossref]
  32. J.-H. Jeon and R. Metzler, “Analysis of short subdiffusive time series: scatter of the time-averaged mean-squared displacement,” J. Phys. A: Math. Theor. 43(25), 252001 (2010).
    [Crossref]
  33. P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
    [Crossref]
  34. F. Stefani, J. Hoogenboom, and E. Barkai, “Beyond quantum jumps: Blinking nanoscale light emitters,” Phys. Today 62(2), 34–39 (2009).
    [Crossref]
  35. J. Cisternas, O. Descalzi, T. Albers, and G. Radons, “Anomalous diffusion of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions,” Phys. Rev. Lett. 116(20), 203901 (2016).
    [Crossref]

2017 (1)

L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I. Lu, V. M. Galitski, and I. B. Spielman, “Brownian motion of solitons in a Bose-Einstein condensate,” Proc. Natl. Acad. Sci. U. S. A. 114(10), 2503–2508 (2017).
[Crossref]

2016 (4)

H. Hennig, T. Neff, and R. Fleischmann, “Dynamical phase diagram of Gaussian wave packets in optical lattices,” Phys. Rev. E 93(3), 032219 (2016).
[Crossref]

Y. Peng, L. Lai, Y.-S. Tai, K. Zhang, X. Xu, and X. Cheng, “Diffusion of ellipsoids in bacterial suspensions,” Phys. Rev. Lett. 116(6), 068303 (2016).
[Crossref]

J. Cisternas, O. Descalzi, T. Albers, and G. Radons, “Anomalous diffusion of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions,” Phys. Rev. Lett. 116(20), 203901 (2016).
[Crossref]

S. V. Dmitriev, E. A. Korznikova, Yu A. Baimova, and M. G. Velarde, “Discrete breathers in crystals,” Phys.-Usp. 59(5), 446–461 (2016).
[Crossref]

2015 (3)

A. Mafi, “Transverse Anderson localization of light: a tutorial,” Adv. Opt. Photonics 7(3), 459–515 (2015).
[Crossref]

Z.-Y. Sun, S. Fishman, and A. Soffer, “Soliton mobility in disordered lattices,” Phys. Rev. E 92(4), 040903 (2015).
[Crossref]

Z.-Y. Sun, S. Fishman, and A. Soffer, “Soliton trapping in a disordered lattice,” Phys. Rev. E 92(1), 012901 (2015).
[Crossref]

2014 (1)

R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, “Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking,” Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014).
[Crossref]

2013 (3)

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7(3), 197–204 (2013).
[Crossref]

U. Naether, M. Heinrich, Y. Lahini, S. Nolte, R. A. Vicencio, M. I. Molina, and A. Szameit, “Self-trapping threshold in disordered nonlinear photonic lattices,” Opt. Lett. 38(9), 1518–1520 (2013).
[Crossref]

V. A. Brazhnyi, C. P. Jisha, and A. S. Rodrigues, “Interaction of discrete nonlinear Schrödinger solitons with a linear lattice impurity,” Phys. Rev. A 87(1), 013609 (2013).
[Crossref]

2011 (1)

R. Franzosi, R. Livi, G. L. Oppo, and A. Politi, “Discrete breathers in Bose-Einstein condensates,” Nonlinearity 24(12), R89–R122 (2011).
[Crossref]

2010 (2)

Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist,” Phys. Rev. E 81(1), 010101 (2010).
[Crossref]

J.-H. Jeon and R. Metzler, “Analysis of short subdiffusive time series: scatter of the time-averaged mean-squared displacement,” J. Phys. A: Math. Theor. 43(25), 252001 (2010).
[Crossref]

2009 (3)

F. Stefani, J. Hoogenboom, and E. Barkai, “Beyond quantum jumps: Blinking nanoscale light emitters,” Phys. Today 62(2), 34–39 (2009).
[Crossref]

K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, “Anderson localization of solitons,” Phys. Rev. Lett. 103(21), 210402 (2009).
[Crossref]

S. Flach, D. O. Krimer, and C. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett. 102(2), 024101 (2009).
[Crossref]

2008 (7)

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett. 100(8), 084103 (2008).
[Crossref]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref]

A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by weak nonlinearity,” Phys. Rev. Lett. 100(9), 094101 (2008).
[Crossref]

S. Flach and A. V. Gorbach, “Discrete breathers–advances in theory and applications,” Phys. Rep. 467(1-3), 1–116 (2008).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Brownian soliton motion,” Phys. Rev. A 77(5), 051802 (2008).
[Crossref]

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
[Crossref]

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys. Rev. Lett. 101(5), 058101 (2008).
[Crossref]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[Crossref]

2006 (1)

I. Golding and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. 96(9), 098102 (2006).
[Crossref]

2005 (1)

Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E 72(2), 026606 (2005).
[Crossref]

2004 (2)

C. Besse, “A relaxation scheme for the nonlinear Schrödinger equation,” SIAM J. Numer. Anal. 42(3), 934–952 (2004).
[Crossref]

D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004).
[Crossref]

1998 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

1993 (1)

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48(4), 3077–3081 (1993).
[Crossref]

1986 (1)

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958).
[Crossref]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Albers, T.

J. Cisternas, O. Descalzi, T. Albers, and G. Radons, “Anomalous diffusion of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions,” Phys. Rev. Lett. 116(20), 203901 (2016).
[Crossref]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958).
[Crossref]

Aubry, S.

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett. 100(8), 084103 (2008).
[Crossref]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref]

Aycock, L. M.

L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I. Lu, V. M. Galitski, and I. B. Spielman, “Brownian motion of solitons in a Bose-Einstein condensate,” Proc. Natl. Acad. Sci. U. S. A. 114(10), 2503–2508 (2017).
[Crossref]

Baimova, Yu A.

S. V. Dmitriev, E. A. Korznikova, Yu A. Baimova, and M. G. Velarde, “Discrete breathers in crystals,” Phys.-Usp. 59(5), 446–461 (2016).
[Crossref]

Barkai, E.

R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, “Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking,” Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014).
[Crossref]

F. Stefani, J. Hoogenboom, and E. Barkai, “Beyond quantum jumps: Blinking nanoscale light emitters,” Phys. Today 62(2), 34–39 (2009).
[Crossref]

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys. Rev. Lett. 101(5), 058101 (2008).
[Crossref]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[Crossref]

Barthelemy, P.

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
[Crossref]

Bertolotti, J.

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature 453(7194), 495–498 (2008).
[Crossref]

Besse, C.

C. Besse, “A relaxation scheme for the nonlinear Schrödinger equation,” SIAM J. Numer. Anal. 42(3), 934–952 (2004).
[Crossref]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Brazhnyi, V. A.

V. A. Brazhnyi, C. P. Jisha, and A. S. Rodrigues, “Interaction of discrete nonlinear Schrödinger solitons with a linear lattice impurity,” Phys. Rev. A 87(1), 013609 (2013).
[Crossref]

Burov, S.

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys. Rev. Lett. 101(5), 058101 (2008).
[Crossref]

Campbell, D. K.

D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004).
[Crossref]

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48(4), 3077–3081 (1993).
[Crossref]

Cheng, X.

Y. Peng, L. Lai, Y.-S. Tai, K. Zhang, X. Xu, and X. Cheng, “Diffusion of ellipsoids in bacterial suspensions,” Phys. Rev. Lett. 116(6), 068303 (2016).
[Crossref]

Cherstvy, A. G.

R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, “Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking,” Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014).
[Crossref]

Christodoulides, D. N.

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7(3), 197–204 (2013).
[Crossref]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref]

Cisternas, J.

J. Cisternas, O. Descalzi, T. Albers, and G. Radons, “Anomalous diffusion of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions,” Phys. Rev. Lett. 116(20), 203901 (2016).
[Crossref]

Cox, E. C.

I. Golding and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. 96(9), 098102 (2006).
[Crossref]

Delande, D.

K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, “Anderson localization of solitons,” Phys. Rev. Lett. 103(21), 210402 (2009).
[Crossref]

Descalzi, O.

J. Cisternas, O. Descalzi, T. Albers, and G. Radons, “Anomalous diffusion of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions,” Phys. Rev. Lett. 116(20), 203901 (2016).
[Crossref]

Dmitriev, S. V.

S. V. Dmitriev, E. A. Korznikova, Yu A. Baimova, and M. G. Velarde, “Discrete breathers in crystals,” Phys.-Usp. 59(5), 446–461 (2016).
[Crossref]

Efimkin, D. K.

L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I. Lu, V. M. Galitski, and I. B. Spielman, “Brownian motion of solitons in a Bose-Einstein condensate,” Proc. Natl. Acad. Sci. U. S. A. 114(10), 2503–2508 (2017).
[Crossref]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Fishman, S.

Z.-Y. Sun, S. Fishman, and A. Soffer, “Soliton trapping in a disordered lattice,” Phys. Rev. E 92(1), 012901 (2015).
[Crossref]

Z.-Y. Sun, S. Fishman, and A. Soffer, “Soliton mobility in disordered lattices,” Phys. Rev. E 92(4), 040903 (2015).
[Crossref]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[Crossref]

Flach, S.

S. Flach, D. O. Krimer, and C. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett. 102(2), 024101 (2009).
[Crossref]

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett. 100(8), 084103 (2008).
[Crossref]

S. Flach and A. V. Gorbach, “Discrete breathers–advances in theory and applications,” Phys. Rep. 467(1-3), 1–116 (2008).
[Crossref]

D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004).
[Crossref]

Fleischmann, R.

H. Hennig, T. Neff, and R. Fleischmann, “Dynamical phase diagram of Gaussian wave packets in optical lattices,” Phys. Rev. E 93(3), 032219 (2016).
[Crossref]

Franzosi, R.

R. Franzosi, R. Livi, G. L. Oppo, and A. Politi, “Discrete breathers in Bose-Einstein condensates,” Nonlinearity 24(12), R89–R122 (2011).
[Crossref]

Galitski, V. M.

L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I. Lu, V. M. Galitski, and I. B. Spielman, “Brownian motion of solitons in a Bose-Einstein condensate,” Proc. Natl. Acad. Sci. U. S. A. 114(10), 2503–2508 (2017).
[Crossref]

Genkina, D.

L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I. Lu, V. M. Galitski, and I. B. Spielman, “Brownian motion of solitons in a Bose-Einstein condensate,” Proc. Natl. Acad. Sci. U. S. A. 114(10), 2503–2508 (2017).
[Crossref]

Golding, I.

I. Golding and E. C. Cox, “Physical nature of bacterial cytoplasm,” Phys. Rev. Lett. 96(9), 098102 (2006).
[Crossref]

Gorbach, A. V.

S. Flach and A. V. Gorbach, “Discrete breathers–advances in theory and applications,” Phys. Rep. 467(1-3), 1–116 (2008).
[Crossref]

Gordon, J. P.

Haus, H. A.

He, Y.

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys. Rev. Lett. 101(5), 058101 (2008).
[Crossref]

Heinrich, M.

Hennig, H.

H. Hennig, T. Neff, and R. Fleischmann, “Dynamical phase diagram of Gaussian wave packets in optical lattices,” Phys. Rev. E 93(3), 032219 (2016).
[Crossref]

Hoogenboom, J.

F. Stefani, J. Hoogenboom, and E. Barkai, “Beyond quantum jumps: Blinking nanoscale light emitters,” Phys. Today 62(2), 34–39 (2009).
[Crossref]

Hurst, H. M.

L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I. Lu, V. M. Galitski, and I. B. Spielman, “Brownian motion of solitons in a Bose-Einstein condensate,” Proc. Natl. Acad. Sci. U. S. A. 114(10), 2503–2508 (2017).
[Crossref]

Jeon, J.-H.

R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, “Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking,” Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014).
[Crossref]

J.-H. Jeon and R. Metzler, “Analysis of short subdiffusive time series: scatter of the time-averaged mean-squared displacement,” J. Phys. A: Math. Theor. 43(25), 252001 (2010).
[Crossref]

Jisha, C. P.

V. A. Brazhnyi, C. P. Jisha, and A. S. Rodrigues, “Interaction of discrete nonlinear Schrödinger solitons with a linear lattice impurity,” Phys. Rev. A 87(1), 013609 (2013).
[Crossref]

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Brownian soliton motion,” Phys. Rev. A 77(5), 051802 (2008).
[Crossref]

Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E 72(2), 026606 (2005).
[Crossref]

Kivshar, Y. S.

D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004).
[Crossref]

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48(4), 3077–3081 (1993).
[Crossref]

Klafter, J.

Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist,” Phys. Rev. E 81(1), 010101 (2010).
[Crossref]

Komineas, S.

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett. 100(8), 084103 (2008).
[Crossref]

Kopidakis, G.

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett. 100(8), 084103 (2008).
[Crossref]

Korznikova, E. A.

S. V. Dmitriev, E. A. Korznikova, Yu A. Baimova, and M. G. Velarde, “Discrete breathers in crystals,” Phys.-Usp. 59(5), 446–461 (2016).
[Crossref]

Krimer, D. O.

S. Flach, D. O. Krimer, and C. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett. 102(2), 024101 (2009).
[Crossref]

Lahini, Y.

U. Naether, M. Heinrich, Y. Lahini, S. Nolte, R. A. Vicencio, M. I. Molina, and A. Szameit, “Self-trapping threshold in disordered nonlinear photonic lattices,” Opt. Lett. 38(9), 1518–1520 (2013).
[Crossref]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref]

Lai, L.

Y. Peng, L. Lai, Y.-S. Tai, K. Zhang, X. Xu, and X. Cheng, “Diffusion of ellipsoids in bacterial suspensions,” Phys. Rev. Lett. 116(6), 068303 (2016).
[Crossref]

Livi, R.

R. Franzosi, R. Livi, G. L. Oppo, and A. Politi, “Discrete breathers in Bose-Einstein condensates,” Nonlinearity 24(12), R89–R122 (2011).
[Crossref]

Lu, H.-I.

L. M. Aycock, H. M. Hurst, D. K. Efimkin, D. Genkina, H.-I. Lu, V. M. Galitski, and I. B. Spielman, “Brownian motion of solitons in a Bose-Einstein condensate,” Proc. Natl. Acad. Sci. U. S. A. 114(10), 2503–2508 (2017).
[Crossref]

Mafi, A.

A. Mafi, “Transverse Anderson localization of light: a tutorial,” Adv. Opt. Photonics 7(3), 459–515 (2015).
[Crossref]

Meroz, Y.

Y. Meroz, I. M. Sokolov, and J. Klafter, “Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist,” Phys. Rev. E 81(1), 010101 (2010).
[Crossref]

Metzler, R.

R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, “Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking,” Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014).
[Crossref]

J.-H. Jeon and R. Metzler, “Analysis of short subdiffusive time series: scatter of the time-averaged mean-squared displacement,” J. Phys. A: Math. Theor. 43(25), 252001 (2010).
[Crossref]

Y. He, S. Burov, R. Metzler, and E. Barkai, “Random time-scale invariant diffusion and transport coefficients,” Phys. Rev. Lett. 101(5), 058101 (2008).
[Crossref]

Molina, M. I.

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[Crossref]

Müller, C. A.

K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, “Anderson localization of solitons,” Phys. Rev. Lett. 103(21), 210402 (2009).
[Crossref]

Naether, U.

Neff, T.

H. Hennig, T. Neff, and R. Fleischmann, “Dynamical phase diagram of Gaussian wave packets in optical lattices,” Phys. Rev. E 93(3), 032219 (2016).
[Crossref]

Nolte, S.

Oppo, G. L.

R. Franzosi, R. Livi, G. L. Oppo, and A. Politi, “Discrete breathers in Bose-Einstein condensates,” Nonlinearity 24(12), R89–R122 (2011).
[Crossref]

Peng, Y.

Y. Peng, L. Lai, Y.-S. Tai, K. Zhang, X. Xu, and X. Cheng, “Diffusion of ellipsoids in bacterial suspensions,” Phys. Rev. Lett. 116(6), 068303 (2016).
[Crossref]

Pikovsky, A. S.

A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by weak nonlinearity,” Phys. Rev. Lett. 100(9), 094101 (2008).
[Crossref]

Politi, A.

R. Franzosi, R. Livi, G. L. Oppo, and A. Politi, “Discrete breathers in Bose-Einstein condensates,” Nonlinearity 24(12), R89–R122 (2011).
[Crossref]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref]

Radons, G.

J. Cisternas, O. Descalzi, T. Albers, and G. Radons, “Anomalous diffusion of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions,” Phys. Rev. Lett. 116(20), 203901 (2016).
[Crossref]

Rodrigues, A. S.

V. A. Brazhnyi, C. P. Jisha, and A. S. Rodrigues, “Interaction of discrete nonlinear Schrödinger solitons with a linear lattice impurity,” Phys. Rev. A 87(1), 013609 (2013).
[Crossref]

Sacha, K.

K. Sacha, C. A. Müller, D. Delande, and J. Zakrzewski, “Anderson localization of solitons,” Phys. Rev. Lett. 103(21), 210402 (2009).
[Crossref]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[Crossref]

Segev, M.

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7(3), 197–204 (2013).
[Crossref]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[Crossref]

Shepelyansky, D. L.

A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by weak nonlinearity,” Phys. Rev. Lett. 100(9), 094101 (2008).
[Crossref]

Silberberg, Y.

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7(3), 197–204 (2013).
[Crossref]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref]

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Other (1)

We address a strong nonlinearity with the ratio of the nonlinearity coefficient to the standard deviation being $500$500 ($\nu /\sigma =500$ν/σ=500). In contrast, previous studies considering the wave packet spreading by weak nonlinearity employed the ratio usually much less than $10$10, even not exceeding $30$30 as for the self-trapping phenomena [13–15]. The strong nonlinearity ensures that the DBSs keep their identity and highly localized no shorter than $z=10^{5}$z=105.

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Figures (3)

Fig. 1.
Fig. 1. (a) Evolution of the DBS in the lattice for a specific realization of disorder; (b) Center of mass of the DBSs $x(z)$ for 1024 independent realizations of disorder. The parameters are $\mu =0.50$, $\sigma =0.002$, and $\nu =1$.
Fig. 2.
Fig. 2. (a) The eMSD $\langle x^{2}(z)\rangle$ as a function of $z$; (b) The tMSDs $\overline {\delta ^{2}(\Delta )}$ (red thin curves) for individual trajectories as functions of $\Delta$, and the trajectory-averaged tMSD $\left \langle \overline {\delta ^{2}(\Delta )} \right \rangle$ is denoted by the black bold curve. All these quantities are evaluated from $N_r=1024$ independent realizations. The parameters are the same as in Fig. 1 ($L=10^{5}$).
Fig. 3.
Fig. 3. Scatter distribution $\phi (\xi )$ for the DBS’s diffusive process. Panels (a)-(c) are extracted at $\Delta =20$, $\Delta =450$, and $\Delta =1250$, corresponding to the diffusion intervals with different exponents $\alpha$. The parameters are the same as in Fig. 2.

Tables (1)

Tables Icon

Table 1. Diffusive property of the DBS’s center of mass. For different intervals of z , α and α are obtained by respectively fitting the eMSD and tMSD curves to x 2 ( z ) z α and δ 2 ( Δ ) ¯ Δ α . The R 2 > 0.997 is kept for every fit.

Equations (5)

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i ψ n z = ( ψ n 1 + ψ n + 1 ) ν | ψ n | 2 ψ n + ϵ n ψ n   ,
x 2 ( z ) 1 N r i = 1 N r x i 2 ( z )   .
δ 2 ( Δ , L ) ¯ 1 L Δ 0 L Δ [ x ( z + Δ ) x ( z ) ] 2 d z   ,
δ 2 ( Δ ) ¯ x 2 ( Δ )   ,
ξ δ 2 ( Δ , L ) ¯ δ 2 ( Δ , L ) ¯   .

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