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

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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]

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]

Skokos, C.

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]

Soffer, A.

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]

Sokolov, I. M.

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]

Sorel, M.

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]

Spielman, I. B.

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]

Stefani, F.

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

Sun, Z.-Y.

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]

Szameit, A.

Tai, Y.-S.

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]

Torner, L.

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

Velarde, M. G.

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]

Vicencio, R. A.

Vysloukh, V. A.

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]

Wiersma, D. S.

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

Xu, 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]

Zakrzewski, J.

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

Zhang, K.

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]

Adv. Opt. Photonics (1)

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

J. Phys. A: Math. Theor. (1)

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]

Nat. Photonics (1)

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

Nature (2)

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

Nonlinearity (1)

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

Opt. Lett. (2)

Phys. Chem. Chem. Phys. (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]

Phys. Rep. (1)

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

Phys. Rev. (1)

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

Phys. Rev. A (2)

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]

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

Phys. Rev. E (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]

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]

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

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

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

Phys. Rev. Lett. (10)

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]

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

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

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]

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]

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, 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]

Phys. Today (2)

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

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

Phys.-Usp. (1)

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]

Proc. Natl. Acad. Sci. U. S. A. (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]

SIAM J. Numer. Anal. (1)

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

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 $z$, $\alpha$ 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.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

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 ) ¯   .