Abstract

We investigate the spatio-temporal dynamics of a ring cavity filled with a non-instantaneous Kerr medium and driven by a coherent injected beam. We show the existence of a stable mixed-mode solution that can be either extended or localized in space. The mixed-mode solutions are obtained in a regime where Turing instability (often called modulational instability) interacts with self-pulsing phenomenon (Andronov-Hopf bifurcation). We numerically describe the transition from stationary inhomogeneous solutions to a branch of mixed-mode solutions. We characterize this transition by constructing the bifurcation diagram associated with these solutions. Finally, we show stable localized mixed-mode solutions, which consist of time-periodic oscillations that are localized in space.

© 2017 Optical Society of America

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References

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  1. M. Tlidi, M. Taki, and T. Kolokolnikov, “Introduction: dissipative localized structures in extended systems,” Chaos 17, 037101 (2007).
    [Crossref] [PubMed]
  2. N. Akhmediev and A. Ankiewicz, (eds.), Dissipative Solitons: from Optics to Biology and Medicine (Lecture Notes in Physics, volume 751, 2008).
  3. H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximatio,” Phy. Rep. 523, 61 (2013).
    [Crossref]
  4. M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimiorv, and M. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
    [Crossref] [PubMed]
  5. M. Tlidi and M.G. Clerc, (eds.), Nonlinear Dynamics: Materials, Theory and Experiments (Springer Proceedings in Physics, 173, 2016).
  6. H. Kidachi, “On mode interactions in reaction diffusion equation with nearly degenerate bifurcations,” Prog. Theor. Phys. 63, 1152–1169 (1980).
    [Crossref]
  7. A.B. Rovinsky and M. Menzinger, “Interaction of Turing and Hopf bifurcations in chemical systems,” Phys. Rev. A 46, 6315 (1992).
    [Crossref] [PubMed]
  8. J.-J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel, and P. Borckmans, “One-dimensional spirals: Novel asynchronous chemical wave sources,” Phys. Rev. Lett. 71, 1272 (1993).
    [Crossref] [PubMed]
  9. G. Heidemann, M. Bode, and H.-G. Purwins, “Fronts between Hopf- and Turing-type domains in a two-component reaction-diffusion system,” Phys. Lett. A 177, 225 (1993).
    [Crossref]
  10. A. Barsella, C. Lepers, M. Taki, and M. Tlidi, “Moving localized structures in quadratic media with a saturable absorber,” Opt. Commun. 232, 381–389 (2004).
    [Crossref]
  11. M. Tlidi, M. Taki, M. Le Berre, E. Reyssayre, A. Tallet, and L. Di Menza, “Moving localized structures and spatial patterns in quadratic media with a saturable absorber,” J. Opt. B: Quantum and Semiclassical Optics 6, S421 (2004).
    [Crossref]
  12. M. Tlidi and M. Haelterman, “Robust Hopf-Turing mixed-mode in optical frequency conversion systems,” Phy. Lett. A 239, 59 (1998).
    [Crossref]
  13. M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524 (1997).
    [Crossref]
  14. A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatiotemporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261 (1996).
    [Crossref]
  15. W.J. Firth, “Spatial instabilities in a Kerr medium with single feedback mirror,” J. Mod. Opt. 37, 151 (1990).
    [Crossref]
  16. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209 (1987).
    [Crossref] [PubMed]
  17. P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phy. Rev. A 48, 4605 (2006).
  18. E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problem (Springer-Verlag, 1993).
  19. M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640 (1994).
    [Crossref] [PubMed]
  20. A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, and L.A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals,  4, 1323 (1994).
    [Crossref]
  21. V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys. 13, 113026 (2011).
    [Crossref]
  22. F. Léo, L. Gelens, Ph. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express 21, 9180–9191 (2013).
    [Crossref] [PubMed]

2014 (1)

M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimiorv, and M. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

2013 (2)

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximatio,” Phy. Rep. 523, 61 (2013).
[Crossref]

F. Léo, L. Gelens, Ph. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express 21, 9180–9191 (2013).
[Crossref] [PubMed]

2011 (1)

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys. 13, 113026 (2011).
[Crossref]

2007 (1)

M. Tlidi, M. Taki, and T. Kolokolnikov, “Introduction: dissipative localized structures in extended systems,” Chaos 17, 037101 (2007).
[Crossref] [PubMed]

2006 (1)

P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phy. Rev. A 48, 4605 (2006).

2004 (2)

A. Barsella, C. Lepers, M. Taki, and M. Tlidi, “Moving localized structures in quadratic media with a saturable absorber,” Opt. Commun. 232, 381–389 (2004).
[Crossref]

M. Tlidi, M. Taki, M. Le Berre, E. Reyssayre, A. Tallet, and L. Di Menza, “Moving localized structures and spatial patterns in quadratic media with a saturable absorber,” J. Opt. B: Quantum and Semiclassical Optics 6, S421 (2004).
[Crossref]

1998 (1)

M. Tlidi and M. Haelterman, “Robust Hopf-Turing mixed-mode in optical frequency conversion systems,” Phy. Lett. A 239, 59 (1998).
[Crossref]

1997 (1)

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524 (1997).
[Crossref]

1996 (1)

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatiotemporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261 (1996).
[Crossref]

1994 (2)

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640 (1994).
[Crossref] [PubMed]

A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, and L.A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals,  4, 1323 (1994).
[Crossref]

1993 (2)

J.-J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel, and P. Borckmans, “One-dimensional spirals: Novel asynchronous chemical wave sources,” Phys. Rev. Lett. 71, 1272 (1993).
[Crossref] [PubMed]

G. Heidemann, M. Bode, and H.-G. Purwins, “Fronts between Hopf- and Turing-type domains in a two-component reaction-diffusion system,” Phys. Lett. A 177, 225 (1993).
[Crossref]

1992 (1)

A.B. Rovinsky and M. Menzinger, “Interaction of Turing and Hopf bifurcations in chemical systems,” Phys. Rev. A 46, 6315 (1992).
[Crossref] [PubMed]

1990 (1)

W.J. Firth, “Spatial instabilities in a Kerr medium with single feedback mirror,” J. Mod. Opt. 37, 151 (1990).
[Crossref]

1987 (1)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209 (1987).
[Crossref] [PubMed]

1980 (1)

H. Kidachi, “On mode interactions in reaction diffusion equation with nearly degenerate bifurcations,” Prog. Theor. Phys. 63, 1152–1169 (1980).
[Crossref]

Barsella, A.

A. Barsella, C. Lepers, M. Taki, and M. Tlidi, “Moving localized structures in quadratic media with a saturable absorber,” Opt. Commun. 232, 381–389 (2004).
[Crossref]

Bode, M.

G. Heidemann, M. Bode, and H.-G. Purwins, “Fronts between Hopf- and Turing-type domains in a two-component reaction-diffusion system,” Phys. Lett. A 177, 225 (1993).
[Crossref]

Borckmans, P.

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatiotemporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261 (1996).
[Crossref]

J.-J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel, and P. Borckmans, “One-dimensional spirals: Novel asynchronous chemical wave sources,” Phys. Rev. Lett. 71, 1272 (1993).
[Crossref] [PubMed]

Clerc, M.

M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimiorv, and M. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Coen, S.

De Kepper, P.

J.-J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel, and P. Borckmans, “One-dimensional spirals: Novel asynchronous chemical wave sources,” Phys. Rev. Lett. 71, 1272 (1993).
[Crossref] [PubMed]

De Wit, A.

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatiotemporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261 (1996).
[Crossref]

J.-J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel, and P. Borckmans, “One-dimensional spirals: Novel asynchronous chemical wave sources,” Phys. Rev. Lett. 71, 1272 (1993).
[Crossref] [PubMed]

Dewel, G.

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatiotemporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261 (1996).
[Crossref]

J.-J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel, and P. Borckmans, “One-dimensional spirals: Novel asynchronous chemical wave sources,” Phys. Rev. Lett. 71, 1272 (1993).
[Crossref] [PubMed]

Di Menza, L.

M. Tlidi, M. Taki, M. Le Berre, E. Reyssayre, A. Tallet, and L. Di Menza, “Moving localized structures and spatial patterns in quadratic media with a saturable absorber,” J. Opt. B: Quantum and Semiclassical Optics 6, S421 (2004).
[Crossref]

Dulos, E.

J.-J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel, and P. Borckmans, “One-dimensional spirals: Novel asynchronous chemical wave sources,” Phys. Rev. Lett. 71, 1272 (1993).
[Crossref] [PubMed]

Emplit, Ph.

Firth, W.J.

A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, and L.A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals,  4, 1323 (1994).
[Crossref]

W.J. Firth, “Spatial instabilities in a Kerr medium with single feedback mirror,” J. Mod. Opt. 37, 151 (1990).
[Crossref]

Gelens, L.

Haelterman, M.

F. Léo, L. Gelens, Ph. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express 21, 9180–9191 (2013).
[Crossref] [PubMed]

M. Tlidi and M. Haelterman, “Robust Hopf-Turing mixed-mode in optical frequency conversion systems,” Phy. Lett. A 239, 59 (1998).
[Crossref]

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524 (1997).
[Crossref]

Hairer, E.

E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problem (Springer-Verlag, 1993).

Heidemann, G.

G. Heidemann, M. Bode, and H.-G. Purwins, “Fronts between Hopf- and Turing-type domains in a two-component reaction-diffusion system,” Phys. Lett. A 177, 225 (1993).
[Crossref]

Kidachi, H.

H. Kidachi, “On mode interactions in reaction diffusion equation with nearly degenerate bifurcations,” Prog. Theor. Phys. 63, 1152–1169 (1980).
[Crossref]

Kockaert, P.

P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phy. Rev. A 48, 4605 (2006).

Kolokolnikov, T.

M. Tlidi, M. Taki, and T. Kolokolnikov, “Introduction: dissipative localized structures in extended systems,” Chaos 17, 037101 (2007).
[Crossref] [PubMed]

Le Berre, M.

M. Tlidi, M. Taki, M. Le Berre, E. Reyssayre, A. Tallet, and L. Di Menza, “Moving localized structures and spatial patterns in quadratic media with a saturable absorber,” J. Opt. B: Quantum and Semiclassical Optics 6, S421 (2004).
[Crossref]

Leblond, H.

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximatio,” Phy. Rep. 523, 61 (2013).
[Crossref]

Lefever, R.

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640 (1994).
[Crossref] [PubMed]

A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, and L.A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals,  4, 1323 (1994).
[Crossref]

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209 (1987).
[Crossref] [PubMed]

Léo, F.

Lepers, C.

A. Barsella, C. Lepers, M. Taki, and M. Tlidi, “Moving localized structures in quadratic media with a saturable absorber,” Opt. Commun. 232, 381–389 (2004).
[Crossref]

Lima, D.

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatiotemporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261 (1996).
[Crossref]

Louvergneaux, E.

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys. 13, 113026 (2011).
[Crossref]

Lugiato, L. A.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209 (1987).
[Crossref] [PubMed]

Lugiato, L.A.

A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, and L.A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals,  4, 1323 (1994).
[Crossref]

Mandel, P.

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524 (1997).
[Crossref]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640 (1994).
[Crossref] [PubMed]

McDonald, G.S.

A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, and L.A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals,  4, 1323 (1994).
[Crossref]

Menzinger, M.

A.B. Rovinsky and M. Menzinger, “Interaction of Turing and Hopf bifurcations in chemical systems,” Phys. Rev. A 46, 6315 (1992).
[Crossref] [PubMed]

Mihalache, D.

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximatio,” Phy. Rep. 523, 61 (2013).
[Crossref]

Nørsett, S.P.

E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problem (Springer-Verlag, 1993).

Odent, V.

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys. 13, 113026 (2011).
[Crossref]

Panajotov, K.

M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimiorv, and M. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Perraud, J.-J.

J.-J. Perraud, A. De Wit, E. Dulos, P. De Kepper, G. Dewel, and P. Borckmans, “One-dimensional spirals: Novel asynchronous chemical wave sources,” Phys. Rev. Lett. 71, 1272 (1993).
[Crossref] [PubMed]

Purwins, H.-G.

G. Heidemann, M. Bode, and H.-G. Purwins, “Fronts between Hopf- and Turing-type domains in a two-component reaction-diffusion system,” Phys. Lett. A 177, 225 (1993).
[Crossref]

Reyssayre, E.

M. Tlidi, M. Taki, M. Le Berre, E. Reyssayre, A. Tallet, and L. Di Menza, “Moving localized structures and spatial patterns in quadratic media with a saturable absorber,” J. Opt. B: Quantum and Semiclassical Optics 6, S421 (2004).
[Crossref]

Rovinsky, A.B.

A.B. Rovinsky and M. Menzinger, “Interaction of Turing and Hopf bifurcations in chemical systems,” Phys. Rev. A 46, 6315 (1992).
[Crossref] [PubMed]

Scorggie, A.J.

A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, and L.A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals,  4, 1323 (1994).
[Crossref]

Staliunas, K.

M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimiorv, and M. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Taki, M.

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys. 13, 113026 (2011).
[Crossref]

M. Tlidi, M. Taki, and T. Kolokolnikov, “Introduction: dissipative localized structures in extended systems,” Chaos 17, 037101 (2007).
[Crossref] [PubMed]

M. Tlidi, M. Taki, M. Le Berre, E. Reyssayre, A. Tallet, and L. Di Menza, “Moving localized structures and spatial patterns in quadratic media with a saturable absorber,” J. Opt. B: Quantum and Semiclassical Optics 6, S421 (2004).
[Crossref]

A. Barsella, C. Lepers, M. Taki, and M. Tlidi, “Moving localized structures in quadratic media with a saturable absorber,” Opt. Commun. 232, 381–389 (2004).
[Crossref]

Tallet, A.

M. Tlidi, M. Taki, M. Le Berre, E. Reyssayre, A. Tallet, and L. Di Menza, “Moving localized structures and spatial patterns in quadratic media with a saturable absorber,” J. Opt. B: Quantum and Semiclassical Optics 6, S421 (2004).
[Crossref]

Tassin, P.

P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phy. Rev. A 48, 4605 (2006).

Tlidi, M.

M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimiorv, and M. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

M. Tlidi, M. Taki, and T. Kolokolnikov, “Introduction: dissipative localized structures in extended systems,” Chaos 17, 037101 (2007).
[Crossref] [PubMed]

P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phy. Rev. A 48, 4605 (2006).

A. Barsella, C. Lepers, M. Taki, and M. Tlidi, “Moving localized structures in quadratic media with a saturable absorber,” Opt. Commun. 232, 381–389 (2004).
[Crossref]

M. Tlidi, M. Taki, M. Le Berre, E. Reyssayre, A. Tallet, and L. Di Menza, “Moving localized structures and spatial patterns in quadratic media with a saturable absorber,” J. Opt. B: Quantum and Semiclassical Optics 6, S421 (2004).
[Crossref]

M. Tlidi and M. Haelterman, “Robust Hopf-Turing mixed-mode in optical frequency conversion systems,” Phy. Lett. A 239, 59 (1998).
[Crossref]

M. Tlidi, P. Mandel, and M. Haelterman, “Spatiotemporal patterns and localized structures in nonlinear optics,” Phys. Rev. E 56, 6524 (1997).
[Crossref]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640 (1994).
[Crossref] [PubMed]

A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, and L.A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals,  4, 1323 (1994).
[Crossref]

Van der Sande, G.

P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phy. Rev. A 48, 4605 (2006).

Veretennicoff, I.

P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phy. Rev. A 48, 4605 (2006).

Vladimiorv, A.G.

M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimiorv, and M. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Wanner, G.

E. Hairer, S.P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problem (Springer-Verlag, 1993).

Chaos (1)

M. Tlidi, M. Taki, and T. Kolokolnikov, “Introduction: dissipative localized structures in extended systems,” Chaos 17, 037101 (2007).
[Crossref] [PubMed]

Chaos, Solitons & Fractals (1)

A.J. Scorggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, and L.A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals,  4, 1323 (1994).
[Crossref]

J. Mod. Opt. (1)

W.J. Firth, “Spatial instabilities in a Kerr medium with single feedback mirror,” J. Mod. Opt. 37, 151 (1990).
[Crossref]

J. Opt. B: Quantum and Semiclassical Optics (1)

M. Tlidi, M. Taki, M. Le Berre, E. Reyssayre, A. Tallet, and L. Di Menza, “Moving localized structures and spatial patterns in quadratic media with a saturable absorber,” J. Opt. B: Quantum and Semiclassical Optics 6, S421 (2004).
[Crossref]

New J. Phys. (1)

V. Odent, M. Taki, and E. Louvergneaux, “Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity,” New J. Phys. 13, 113026 (2011).
[Crossref]

Opt. Commun. (1)

A. Barsella, C. Lepers, M. Taki, and M. Tlidi, “Moving localized structures in quadratic media with a saturable absorber,” Opt. Commun. 232, 381–389 (2004).
[Crossref]

Opt. Express (1)

Phil. Trans. R. Soc. A (1)

M. Tlidi, K. Staliunas, K. Panajotov, A.G. Vladimiorv, and M. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Phil. Trans. R. Soc. A 372, 20140101 (2014).
[Crossref] [PubMed]

Phy. Lett. A (1)

M. Tlidi and M. Haelterman, “Robust Hopf-Turing mixed-mode in optical frequency conversion systems,” Phy. Lett. A 239, 59 (1998).
[Crossref]

Phy. Rep. (1)

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[Crossref]

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P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phy. Rev. A 48, 4605 (2006).

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[Crossref]

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[Crossref]

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[Crossref]

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M. Tlidi and M.G. Clerc, (eds.), Nonlinear Dynamics: Materials, Theory and Experiments (Springer Proceedings in Physics, 173, 2016).

N. Akhmediev and A. Ankiewicz, (eds.), Dissipative Solitons: from Optics to Biology and Medicine (Lecture Notes in Physics, volume 751, 2008).

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Figures (5)

Fig. 1
Fig. 1 Marginal stability curves. The parameters are γ = 1, δ = 1 with IH and IT are Hopf and Turing instability thresholds, respectively. (a) d = 0.4, above the upper curve HSS is Hopf unstable. Inside the lower curve, HSS is Turing unstable. (b) d = 0.81, note the co-dimension two instability where Turing and Hopf thresholds coincide. (c) d = 0.85, the situation is inverted with respect to (a).
Fig. 2
Fig. 2 Space-time map showing the formation of a Turing-Hopf mixed-mode solution. The parameters are γ = 1, δ = 1, S = 5.28 and d = 0.4.
Fig. 3
Fig. 3 Bifurcation diagram showing the evolution of the intracavity field intensity as a function of the input field amplitude. Stable (unstable) homogeneous steady states are indicated by solid (dotted) line. Filled squares denote the maximum intensity of the stationary periodic structures. Empty squares indicate the maximum intensity associated with mixed-mode solutions. Parameters are γ = 1, δ = 1, and d = 0.4.
Fig. 4
Fig. 4 Space-time maps displaying the formation of multi-peaks stationary localized structures. (a) one, (b) two and (c) three peaks obtained by numerical simulations of Eqs. (1) and (2) for the parameters γ = 1.3, δ = 3, S = 1.9, and d = 1.45.
Fig. 5
Fig. 5 (a) Space-time map showing the generation of a localized mixed-mode solution. Parameters are γ = 1.3, δ = 3, S = 2.11, and d = 1.45. (b) Bifurcation diagram showing the evolution of the intracavity field intensity as a function of the input field amplitude. Stable (unstable) homogeneous steady states are indicated by solid (dotted) line. Filled squares denote the maximum intensity of the stationary localized structures. Empty squares indicate the maximum intensity associated with localized mixed-mode solutions. Same parameters as in (a) with varying S.

Equations (2)

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E t = S ( 1 + i δ ) E + i n E + i 2 E x 2 ,
γ n t = n + | E | 2 + d 2 n x 2 ,

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