Abstract

Driven optical systems can exhibit coexistence of equilibrium states. Traveling waves or fronts between different states present complex spatiotemporal dynamics. We investigate the mechanisms that govern the front spread. Based on a liquid crystal light valve experiment with optical feedback, we show that the front propagation does not pursue a minimization of free energy. Depending on the free propagation length in the optical feedback loop, the front speed exhibits a supercritical transition. Theoretically, from first principles, we use a model that takes it into account, characterizing the speed transition from a plateau to a growing regime. The theoretical and experimental results show quite fair agreement.

© 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. W. van Saarloos, “Front propagation into unstable states,” Phys. Reports 386(2), 29–222 (2003).
    [Crossref]
  2. R. A. Fisher, “The wave of advance of advantageous genes,” Ann. Eugen. 7, 353–369 (1937).
    [Crossref]
  3. A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov., “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem,” Bull. Mosc. Univ. Math. Mech.1(6), 1–26 (1937). Reprinted in: V. M. Tikhomirov (ed.), Selected Works of A. N. Kolmogorov, vol. 1 (Kluwer, 1991). Also in: O. A. Oleinik, I. G. Petrowsky Selected Works, Part II (Gordon and Breach, 1996).
  4. M. G. Clerc, S. Residori, and C. S Riera, “First-order Fréedericksz transition in the presence of light-driven feedback in nematic liquid crystals,” Phys. Rev. E 63, 060701 (2001).
    [Crossref]
  5. S. Residori, A. Petrossian, T. Nagaya, C. S. Riera, and M. G. Clerc, “Fronts and localized structures in a liquid crystal-light-valve with optical feedback,” Phys. D: Nonlinear Phenom. 199, 149–165 (2004).
    [Crossref]
  6. S. Residori, “Patterns, fronts and structures in a liquid crystal-Light-Valve with optical feedback,” Phys. Reports 416, 201–272 (2005).
    [Crossref]
  7. M. G. Clerc, A. Petrossian, and S. Residori, “Bouncing localized structures in a liquid crystal light-valve experiment,” Phys. Rev. E 71, 015205 (2005).
    [Crossref]
  8. M. G. Clerc, T. Nagaya, A. Petrossian, S. Residori, and C. S. Riera, “First-order Frédericksz transition and front propagation in a liquid crystal light valve with feedback,” Euro Phys. J. D 28(3), 435–445 (2004).
    [Crossref]
  9. F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Driven Front Propagation in 1D Spatially Periodic Media,” Phys. Rev. Lett. 103, 128003 (2009).
    [Crossref] [PubMed]
  10. F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Front dynamics and pinning-depinning phenomenon in spatially periodic media,” Phys. Rev. E 81, 056203 (2010).
    [Crossref]
  11. A. J. Álvarez-Socorro, M. G. Clerc, G. González-Cortés, and M. Wilson, “Nonvariational mechanism of front propagation: Theory and experiments,” Phys. Rev. E 95, 010202 (2017).
    [Crossref] [PubMed]
  12. K. Alfaro-Bittner, M. G. Clerc, M. García-Nustes, and R. G. Rojas, “π-kink propagation in the damped Frenkel-Kontorova model,” Europhys. Lett. 119(4), 40003 (2018).
    [Crossref]
  13. I. Prigogine and G. Nicolis, Self-Organization in Non-Equilibrium Systems, (John Wiley & Sons Inc., 1997).
  14. H. Haken, Advanced synergetics: Instability hierarchies of self-organizing systems and devices, (Springer Verlag, 1983).
  15. M. G. Clerc, G. González-Cortés, V. Odent, and M. Wilson, “Optical textures: characterizing spatiotemporal chaos,” Opt. Express 24(14), 15478–15485 (2016).
    [Crossref] [PubMed]
  16. K. P. Hadeler and F. Rothe., “Travelling fronts in nonlinear diffusion equations,” J. Math. Biol. 2(3), 251–263 (1975).
    [Crossref]
  17. E. I. Fredholm, “Sur une classe d’equations fonctionnelles,” Acta Math. 27, 365–390 (1903).
    [Crossref]
  18. D. Mollison, “Spatial contact models for ecological and epidemic spread,” J. R. Statist. Soc. B 39, 283–326 (1977).

2018 (1)

K. Alfaro-Bittner, M. G. Clerc, M. García-Nustes, and R. G. Rojas, “π-kink propagation in the damped Frenkel-Kontorova model,” Europhys. Lett. 119(4), 40003 (2018).
[Crossref]

2017 (1)

A. J. Álvarez-Socorro, M. G. Clerc, G. González-Cortés, and M. Wilson, “Nonvariational mechanism of front propagation: Theory and experiments,” Phys. Rev. E 95, 010202 (2017).
[Crossref] [PubMed]

2016 (1)

2010 (1)

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Front dynamics and pinning-depinning phenomenon in spatially periodic media,” Phys. Rev. E 81, 056203 (2010).
[Crossref]

2009 (1)

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Driven Front Propagation in 1D Spatially Periodic Media,” Phys. Rev. Lett. 103, 128003 (2009).
[Crossref] [PubMed]

2005 (2)

S. Residori, “Patterns, fronts and structures in a liquid crystal-Light-Valve with optical feedback,” Phys. Reports 416, 201–272 (2005).
[Crossref]

M. G. Clerc, A. Petrossian, and S. Residori, “Bouncing localized structures in a liquid crystal light-valve experiment,” Phys. Rev. E 71, 015205 (2005).
[Crossref]

2004 (2)

M. G. Clerc, T. Nagaya, A. Petrossian, S. Residori, and C. S. Riera, “First-order Frédericksz transition and front propagation in a liquid crystal light valve with feedback,” Euro Phys. J. D 28(3), 435–445 (2004).
[Crossref]

S. Residori, A. Petrossian, T. Nagaya, C. S. Riera, and M. G. Clerc, “Fronts and localized structures in a liquid crystal-light-valve with optical feedback,” Phys. D: Nonlinear Phenom. 199, 149–165 (2004).
[Crossref]

2003 (1)

W. van Saarloos, “Front propagation into unstable states,” Phys. Reports 386(2), 29–222 (2003).
[Crossref]

2001 (1)

M. G. Clerc, S. Residori, and C. S Riera, “First-order Fréedericksz transition in the presence of light-driven feedback in nematic liquid crystals,” Phys. Rev. E 63, 060701 (2001).
[Crossref]

1977 (1)

D. Mollison, “Spatial contact models for ecological and epidemic spread,” J. R. Statist. Soc. B 39, 283–326 (1977).

1975 (1)

K. P. Hadeler and F. Rothe., “Travelling fronts in nonlinear diffusion equations,” J. Math. Biol. 2(3), 251–263 (1975).
[Crossref]

1937 (1)

R. A. Fisher, “The wave of advance of advantageous genes,” Ann. Eugen. 7, 353–369 (1937).
[Crossref]

1903 (1)

E. I. Fredholm, “Sur une classe d’equations fonctionnelles,” Acta Math. 27, 365–390 (1903).
[Crossref]

Alfaro-Bittner, K.

K. Alfaro-Bittner, M. G. Clerc, M. García-Nustes, and R. G. Rojas, “π-kink propagation in the damped Frenkel-Kontorova model,” Europhys. Lett. 119(4), 40003 (2018).
[Crossref]

Álvarez-Socorro, A. J.

A. J. Álvarez-Socorro, M. G. Clerc, G. González-Cortés, and M. Wilson, “Nonvariational mechanism of front propagation: Theory and experiments,” Phys. Rev. E 95, 010202 (2017).
[Crossref] [PubMed]

Bortolozzo, U.

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Front dynamics and pinning-depinning phenomenon in spatially periodic media,” Phys. Rev. E 81, 056203 (2010).
[Crossref]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Driven Front Propagation in 1D Spatially Periodic Media,” Phys. Rev. Lett. 103, 128003 (2009).
[Crossref] [PubMed]

Clerc, M. G.

K. Alfaro-Bittner, M. G. Clerc, M. García-Nustes, and R. G. Rojas, “π-kink propagation in the damped Frenkel-Kontorova model,” Europhys. Lett. 119(4), 40003 (2018).
[Crossref]

A. J. Álvarez-Socorro, M. G. Clerc, G. González-Cortés, and M. Wilson, “Nonvariational mechanism of front propagation: Theory and experiments,” Phys. Rev. E 95, 010202 (2017).
[Crossref] [PubMed]

M. G. Clerc, G. González-Cortés, V. Odent, and M. Wilson, “Optical textures: characterizing spatiotemporal chaos,” Opt. Express 24(14), 15478–15485 (2016).
[Crossref] [PubMed]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Front dynamics and pinning-depinning phenomenon in spatially periodic media,” Phys. Rev. E 81, 056203 (2010).
[Crossref]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Driven Front Propagation in 1D Spatially Periodic Media,” Phys. Rev. Lett. 103, 128003 (2009).
[Crossref] [PubMed]

M. G. Clerc, A. Petrossian, and S. Residori, “Bouncing localized structures in a liquid crystal light-valve experiment,” Phys. Rev. E 71, 015205 (2005).
[Crossref]

S. Residori, A. Petrossian, T. Nagaya, C. S. Riera, and M. G. Clerc, “Fronts and localized structures in a liquid crystal-light-valve with optical feedback,” Phys. D: Nonlinear Phenom. 199, 149–165 (2004).
[Crossref]

M. G. Clerc, T. Nagaya, A. Petrossian, S. Residori, and C. S. Riera, “First-order Frédericksz transition and front propagation in a liquid crystal light valve with feedback,” Euro Phys. J. D 28(3), 435–445 (2004).
[Crossref]

M. G. Clerc, S. Residori, and C. S Riera, “First-order Fréedericksz transition in the presence of light-driven feedback in nematic liquid crystals,” Phys. Rev. E 63, 060701 (2001).
[Crossref]

Elias, R. G.

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Front dynamics and pinning-depinning phenomenon in spatially periodic media,” Phys. Rev. E 81, 056203 (2010).
[Crossref]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Driven Front Propagation in 1D Spatially Periodic Media,” Phys. Rev. Lett. 103, 128003 (2009).
[Crossref] [PubMed]

Fisher, R. A.

R. A. Fisher, “The wave of advance of advantageous genes,” Ann. Eugen. 7, 353–369 (1937).
[Crossref]

Fredholm, E. I.

E. I. Fredholm, “Sur une classe d’equations fonctionnelles,” Acta Math. 27, 365–390 (1903).
[Crossref]

García-Nustes, M.

K. Alfaro-Bittner, M. G. Clerc, M. García-Nustes, and R. G. Rojas, “π-kink propagation in the damped Frenkel-Kontorova model,” Europhys. Lett. 119(4), 40003 (2018).
[Crossref]

González-Cortés, G.

A. J. Álvarez-Socorro, M. G. Clerc, G. González-Cortés, and M. Wilson, “Nonvariational mechanism of front propagation: Theory and experiments,” Phys. Rev. E 95, 010202 (2017).
[Crossref] [PubMed]

M. G. Clerc, G. González-Cortés, V. Odent, and M. Wilson, “Optical textures: characterizing spatiotemporal chaos,” Opt. Express 24(14), 15478–15485 (2016).
[Crossref] [PubMed]

Hadeler, K. P.

K. P. Hadeler and F. Rothe., “Travelling fronts in nonlinear diffusion equations,” J. Math. Biol. 2(3), 251–263 (1975).
[Crossref]

Haken, H.

H. Haken, Advanced synergetics: Instability hierarchies of self-organizing systems and devices, (Springer Verlag, 1983).

Haudin, F.

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Front dynamics and pinning-depinning phenomenon in spatially periodic media,” Phys. Rev. E 81, 056203 (2010).
[Crossref]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Driven Front Propagation in 1D Spatially Periodic Media,” Phys. Rev. Lett. 103, 128003 (2009).
[Crossref] [PubMed]

Kolmogorov, A. N.

A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov., “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem,” Bull. Mosc. Univ. Math. Mech.1(6), 1–26 (1937). Reprinted in: V. M. Tikhomirov (ed.), Selected Works of A. N. Kolmogorov, vol. 1 (Kluwer, 1991). Also in: O. A. Oleinik, I. G. Petrowsky Selected Works, Part II (Gordon and Breach, 1996).

Mollison, D.

D. Mollison, “Spatial contact models for ecological and epidemic spread,” J. R. Statist. Soc. B 39, 283–326 (1977).

Nagaya, T.

M. G. Clerc, T. Nagaya, A. Petrossian, S. Residori, and C. S. Riera, “First-order Frédericksz transition and front propagation in a liquid crystal light valve with feedback,” Euro Phys. J. D 28(3), 435–445 (2004).
[Crossref]

S. Residori, A. Petrossian, T. Nagaya, C. S. Riera, and M. G. Clerc, “Fronts and localized structures in a liquid crystal-light-valve with optical feedback,” Phys. D: Nonlinear Phenom. 199, 149–165 (2004).
[Crossref]

Nicolis, G.

I. Prigogine and G. Nicolis, Self-Organization in Non-Equilibrium Systems, (John Wiley & Sons Inc., 1997).

Odent, V.

Petrossian, A.

M. G. Clerc, A. Petrossian, and S. Residori, “Bouncing localized structures in a liquid crystal light-valve experiment,” Phys. Rev. E 71, 015205 (2005).
[Crossref]

S. Residori, A. Petrossian, T. Nagaya, C. S. Riera, and M. G. Clerc, “Fronts and localized structures in a liquid crystal-light-valve with optical feedback,” Phys. D: Nonlinear Phenom. 199, 149–165 (2004).
[Crossref]

M. G. Clerc, T. Nagaya, A. Petrossian, S. Residori, and C. S. Riera, “First-order Frédericksz transition and front propagation in a liquid crystal light valve with feedback,” Euro Phys. J. D 28(3), 435–445 (2004).
[Crossref]

Petrovskii, I. G.

A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov., “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem,” Bull. Mosc. Univ. Math. Mech.1(6), 1–26 (1937). Reprinted in: V. M. Tikhomirov (ed.), Selected Works of A. N. Kolmogorov, vol. 1 (Kluwer, 1991). Also in: O. A. Oleinik, I. G. Petrowsky Selected Works, Part II (Gordon and Breach, 1996).

Piskunov., N. S.

A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov., “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem,” Bull. Mosc. Univ. Math. Mech.1(6), 1–26 (1937). Reprinted in: V. M. Tikhomirov (ed.), Selected Works of A. N. Kolmogorov, vol. 1 (Kluwer, 1991). Also in: O. A. Oleinik, I. G. Petrowsky Selected Works, Part II (Gordon and Breach, 1996).

Prigogine, I.

I. Prigogine and G. Nicolis, Self-Organization in Non-Equilibrium Systems, (John Wiley & Sons Inc., 1997).

Residori, S.

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Front dynamics and pinning-depinning phenomenon in spatially periodic media,” Phys. Rev. E 81, 056203 (2010).
[Crossref]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Driven Front Propagation in 1D Spatially Periodic Media,” Phys. Rev. Lett. 103, 128003 (2009).
[Crossref] [PubMed]

M. G. Clerc, A. Petrossian, and S. Residori, “Bouncing localized structures in a liquid crystal light-valve experiment,” Phys. Rev. E 71, 015205 (2005).
[Crossref]

S. Residori, “Patterns, fronts and structures in a liquid crystal-Light-Valve with optical feedback,” Phys. Reports 416, 201–272 (2005).
[Crossref]

M. G. Clerc, T. Nagaya, A. Petrossian, S. Residori, and C. S. Riera, “First-order Frédericksz transition and front propagation in a liquid crystal light valve with feedback,” Euro Phys. J. D 28(3), 435–445 (2004).
[Crossref]

S. Residori, A. Petrossian, T. Nagaya, C. S. Riera, and M. G. Clerc, “Fronts and localized structures in a liquid crystal-light-valve with optical feedback,” Phys. D: Nonlinear Phenom. 199, 149–165 (2004).
[Crossref]

M. G. Clerc, S. Residori, and C. S Riera, “First-order Fréedericksz transition in the presence of light-driven feedback in nematic liquid crystals,” Phys. Rev. E 63, 060701 (2001).
[Crossref]

Riera, C. S

M. G. Clerc, S. Residori, and C. S Riera, “First-order Fréedericksz transition in the presence of light-driven feedback in nematic liquid crystals,” Phys. Rev. E 63, 060701 (2001).
[Crossref]

Riera, C. S.

M. G. Clerc, T. Nagaya, A. Petrossian, S. Residori, and C. S. Riera, “First-order Frédericksz transition and front propagation in a liquid crystal light valve with feedback,” Euro Phys. J. D 28(3), 435–445 (2004).
[Crossref]

S. Residori, A. Petrossian, T. Nagaya, C. S. Riera, and M. G. Clerc, “Fronts and localized structures in a liquid crystal-light-valve with optical feedback,” Phys. D: Nonlinear Phenom. 199, 149–165 (2004).
[Crossref]

Rojas, R. G.

K. Alfaro-Bittner, M. G. Clerc, M. García-Nustes, and R. G. Rojas, “π-kink propagation in the damped Frenkel-Kontorova model,” Europhys. Lett. 119(4), 40003 (2018).
[Crossref]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Front dynamics and pinning-depinning phenomenon in spatially periodic media,” Phys. Rev. E 81, 056203 (2010).
[Crossref]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Driven Front Propagation in 1D Spatially Periodic Media,” Phys. Rev. Lett. 103, 128003 (2009).
[Crossref] [PubMed]

Rothe., F.

K. P. Hadeler and F. Rothe., “Travelling fronts in nonlinear diffusion equations,” J. Math. Biol. 2(3), 251–263 (1975).
[Crossref]

van Saarloos, W.

W. van Saarloos, “Front propagation into unstable states,” Phys. Reports 386(2), 29–222 (2003).
[Crossref]

Wilson, M.

A. J. Álvarez-Socorro, M. G. Clerc, G. González-Cortés, and M. Wilson, “Nonvariational mechanism of front propagation: Theory and experiments,” Phys. Rev. E 95, 010202 (2017).
[Crossref] [PubMed]

M. G. Clerc, G. González-Cortés, V. Odent, and M. Wilson, “Optical textures: characterizing spatiotemporal chaos,” Opt. Express 24(14), 15478–15485 (2016).
[Crossref] [PubMed]

Acta Math. (1)

E. I. Fredholm, “Sur une classe d’equations fonctionnelles,” Acta Math. 27, 365–390 (1903).
[Crossref]

Ann. Eugen. (1)

R. A. Fisher, “The wave of advance of advantageous genes,” Ann. Eugen. 7, 353–369 (1937).
[Crossref]

Euro Phys. J. D (1)

M. G. Clerc, T. Nagaya, A. Petrossian, S. Residori, and C. S. Riera, “First-order Frédericksz transition and front propagation in a liquid crystal light valve with feedback,” Euro Phys. J. D 28(3), 435–445 (2004).
[Crossref]

Europhys. Lett. (1)

K. Alfaro-Bittner, M. G. Clerc, M. García-Nustes, and R. G. Rojas, “π-kink propagation in the damped Frenkel-Kontorova model,” Europhys. Lett. 119(4), 40003 (2018).
[Crossref]

J. Math. Biol. (1)

K. P. Hadeler and F. Rothe., “Travelling fronts in nonlinear diffusion equations,” J. Math. Biol. 2(3), 251–263 (1975).
[Crossref]

J. R. Statist. Soc. B (1)

D. Mollison, “Spatial contact models for ecological and epidemic spread,” J. R. Statist. Soc. B 39, 283–326 (1977).

Opt. Express (1)

Phys. D: Nonlinear Phenom. (1)

S. Residori, A. Petrossian, T. Nagaya, C. S. Riera, and M. G. Clerc, “Fronts and localized structures in a liquid crystal-light-valve with optical feedback,” Phys. D: Nonlinear Phenom. 199, 149–165 (2004).
[Crossref]

Phys. Reports (2)

S. Residori, “Patterns, fronts and structures in a liquid crystal-Light-Valve with optical feedback,” Phys. Reports 416, 201–272 (2005).
[Crossref]

W. van Saarloos, “Front propagation into unstable states,” Phys. Reports 386(2), 29–222 (2003).
[Crossref]

Phys. Rev. E (4)

M. G. Clerc, S. Residori, and C. S Riera, “First-order Fréedericksz transition in the presence of light-driven feedback in nematic liquid crystals,” Phys. Rev. E 63, 060701 (2001).
[Crossref]

M. G. Clerc, A. Petrossian, and S. Residori, “Bouncing localized structures in a liquid crystal light-valve experiment,” Phys. Rev. E 71, 015205 (2005).
[Crossref]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Front dynamics and pinning-depinning phenomenon in spatially periodic media,” Phys. Rev. E 81, 056203 (2010).
[Crossref]

A. J. Álvarez-Socorro, M. G. Clerc, G. González-Cortés, and M. Wilson, “Nonvariational mechanism of front propagation: Theory and experiments,” Phys. Rev. E 95, 010202 (2017).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc, and S. Residori, “Driven Front Propagation in 1D Spatially Periodic Media,” Phys. Rev. Lett. 103, 128003 (2009).
[Crossref] [PubMed]

Other (3)

A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov., “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem,” Bull. Mosc. Univ. Math. Mech.1(6), 1–26 (1937). Reprinted in: V. M. Tikhomirov (ed.), Selected Works of A. N. Kolmogorov, vol. 1 (Kluwer, 1991). Also in: O. A. Oleinik, I. G. Petrowsky Selected Works, Part II (Gordon and Breach, 1996).

I. Prigogine and G. Nicolis, Self-Organization in Non-Equilibrium Systems, (John Wiley & Sons Inc., 1997).

H. Haken, Advanced synergetics: Instability hierarchies of self-organizing systems and devices, (Springer Verlag, 1983).

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Figures (5)

Fig. 1
Fig. 1 Liquid crystal light valve with optical feedback. (a) Schematic representation of the experimental setup. The LCLV is composed of a nematic liquid crystal film sandwiched in between a glass and a photoconductive plate-over with a dielectric mirror. The light is injected through a He-Ne laser beam, f stands for lenses with a focal length of 25 cm, PBS represents a polarizer beam splitter, BS a beam splitter, and SLM is a spatial light modulator controlled by a computer (PC). The feedback loop is closed by an optical fiber bundle (FB). The free propagation length is denoted by L and the image in the LCLV is captured through a CCD camera. (b) Temporal snapshots sequence of the front propagation showed in the LCLV taken at L = 0 mm, ν = 1 KHz, and V0 = 2.62 Vrms. Dark and light area account for different average molecular orientations, respectively. The dashed rectangles mark the illuminated region.
Fig. 2
Fig. 2 Experimental characterization of the bifurcation diagram and the front propagation transition. (a) Bifurcation diagram observed in the LCLV with optical feedback constructed at L = 0 mm. The points account for the intensity of the reflected light by the LCLV as a function of the applied voltage V0. The system exhibits three regions, two monostable and one bistable between the planar and reoriented state. VFT accounts for the critical value of the reorientation instability, the Fréedericksz transition. The insets stand for respective snapshots obtained in the indicated voltages. (b) Front speed as a function of free propagation length L at V0 = 2.62Vrms. The points account for the front speed measured in pixels per second. The dashed line is the union between consecutive experimental points. The continuous curve stands for the trend line of the experimental points.
Fig. 3
Fig. 3 Characterization of bifurcation diagram and front speed of model Eq. (1). (a) Bifurcation diagram of Eq. (1). Equilibrium amplitude uo as a function of the parameter μ for fixed β. The continuous and dashed curves account for stable and unstable equilibrium, respectively. These curves were obtained by solving the algebraic equation 0 = μ u 0 + β u 0 2 + u 0 3 u 0 5; up, u, and up account for the upper, middle, and lower equilibrium branch, respectively. The system exhibits three regions, two monostable and one bistable. (b) Front speed as a function of free propagation length L. The continuous curve shows the front speed of model Eq. (1) obtained numerically with μ = 1.0, β = 0.1, and b = c = L. The dashed horizontal curve accounts for the minimal front speed using the marginal criterion v min = 2 μ.
Fig. 4
Fig. 4 Front propagation into an unstable state of model Eq. (1). (a) Spatiotemporal evolution of amplitud of critical model u(x, t) of model Eq. (1) by μ = 1.0, β = 0.1, and b = 0. Temporal evolution of the front propagation after (c = 0, t < 20) and before (c = −30, t ⩾ 20) consider the nonvariational advection term. The arrows show the direction of front propagation in the respective periods. Front profiles at t = 35 (b) and t = 10 (c), respectively.
Fig. 5
Fig. 5 Spatiotemporal propagation of front solution into an unstable state for different free propagation lengths. Top panels account for front propagation in the experiment by L = −0.4 cm (a), L = 0.0 cm (b), and L = 0.4 cm (c), respectively. Bottom panels stand for the front propagation of model Eq. (1) by μ = 1.0, β = 0.1, and free propagation length L = −1.0 (d), L = 0 (e), and L = 4.0 (f). The insets account for the front profile at a given instant experimentally and numerically, respectively.

Equations (6)

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t u = μ u + β u 2 + γ u 3 u 5 + x x u + b u x x u + c ( x u ) 2 ,
t u = u ,
u ( x , t ) = u f ( z x v 0 t P ˙ ( t ) ) + w ( x v 0 t p ( t ) ) ,
w = p ˙ ( t ) z u f b u f z z u f c ( z u f ) 2 ,
p ˙ ( t ) = v n v b ϕ | u f z z u f ϕ | z u f c ϕ | ( z u f ) 2 ϕ | z u f ,
v = v 0 + v n v .

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