October  2011, 4(5): 1341-1357. doi: 10.3934/dcdss.2011.4.1341

Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities

1. 

Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, United Kingdom, United Kingdom

Received  September 2009 Revised  January 2010 Published  December 2010

This paper continues an investigation into a one-dimensional lattice equation that models the light field in a system comprised of a periodic array of pumped optical cavities with saturable nonlinearity. The additional effects of a spatial gradient of the phase of the pump field are studied, which in the presence of loss terms is shown to break the spatial reversibility of the steady problem. Unlike for continuum systems, small symmetry-breaking is argued to not lead directly to moving solitons, but there remains a pinning region in which there are infinitely many distinct stable stationary solitons of arbitrarily large width. These solitons are no-longer arranged in a homoclinic snaking bifurcation diagrams, but instead break up into discrete isolas. For large enough symmetry-breaking, the fold bifurcations of the lowest intensity solitons no longer overlap, which is argued to be the trigger point of moving localised structures. Due to the dissipative nature of the problem, any radiation shed by these structures is damped and so they appear to be true attractors. Careful direct numerical simulations reveal that branches of the moving solitons undergo unsual hysteresis with respect to the pump, for sufficiently large symmetry breaking.
Citation: Alexey Yulin, Alan Champneys. Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1341-1357. doi: 10.3934/dcdss.2011.4.1341
References:
[1]

M. Beck, J. Knobloch, D. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders and isolas of localized patterns,, SIAM J. Math. Anal., 41 (2009), 936. doi: 10.1137/080713306. Google Scholar

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J. Burke and E. Knobloch, Homoclinic snaking: Structure and stability,, Chaos, 17 (2007). doi: 10.1063/1.2746816. Google Scholar

[5]

J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift-Hohenberg equation,, Phys. Lett. A, 360 (2007), 681. doi: 10.1016/j.physleta.2006.08.072. Google Scholar

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P. Coullet, C. Riera and C. Tresser, Stable static localized structures in one dimension,, Phys. Rev. Lett., 84 (2000), 3069. doi: 10.1103/PhysRevLett.84.3069. Google Scholar

[7]

G. Dangelmayr, J. Hettel and E. Knobloch, Parity-breaking bifurcation in inhomogeneous systems, Nonlinearity, 10 (1997), 1093. doi: 10.1088/0951-7715/10/5/006. Google Scholar

[8]

J. H. P. Dawes, Localized pattern formation with a large scale mode: Slanted snaking,, SIAM J. Appl. Dyn. Syst., 7 (2008), 186. doi: 10.1137/06067794X. Google Scholar

[9]

O. A. Egorov, F. Lederer and Y. S. Kivshar, How does an inclined holding beam affect discrete modulational instability and solitons in nonlinear cavities?,, Optics Express, 15 (2007), 4149. doi: 10.1364/OE.15.004149. Google Scholar

[10]

O. Egorov, U. Peschel and F. Lederer, Mobility of discrete cavity solitons,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.066603. Google Scholar

[11]

O. A. Egorov, U. Peschel and F. Lederer, Discrete quadratic cavity solitons,, Phys. Rev. E, 71 (2005). doi: 10.1103/PhysRevE.71.056612. Google Scholar

[12]

O. A. Egorov, D. V. Skryabin, A. V. Yulin and F. Lederer, Bright cavity polariton solitons,, Phys. Rev. Lett., 102 (2009). doi: 10.1103/PhysRevLett.102.153904. Google Scholar

[13]

W. J. Firth and A. J. Scroggie, Optical bullet holes: Robust controllable localized states of a nonlinear cavity,, Phys. Rev. Lett., 76 (1996), 1623. doi: 10.1103/PhysRevLett.76.1623. Google Scholar

[14]

D. Gomilla and G. L. Oppo, Subcritical patterns and dissipative solitons due to intracavity photonic crystals,, Phys. Rev. A, 76 (2007). doi: 10.1103/PhysRevA.76.043823. Google Scholar

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A. Hagberg and E. Meron, Propgation failure in excitable media,, Phys. Rev. E, 57 (1998), 229. doi: 10.1103/PhysRevE.57.299. Google Scholar

[16]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc and S. Residori, Driven front propogation in 1D spatially periodic media,, Phys. Rev. Lett., 103 (2009). doi: 10.1103/PhysRevLett.103.128003. Google Scholar

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J. Knobloch, D. Lloyd, B. Sandstede and T. Wagenknecht, Isolas of two-pulse solutions in homoclinic snaking scenarios,, preprint (2009)., (2009). Google Scholar

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G. Kozyreff and S. J. Chapman, Asymptotics of large bound states of localized structures,, Phys. Rev. Lett., 97 (2006). doi: 10.1103/PhysRevLett.97.044502. Google Scholar

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J. S. W. Lamb and H. W. Capel, Local bifurcations on the plane with reversing point group symmetry,, Chaos, 5 (1995), 271. doi: 10.1016/0960-0779(93)E0022-4. Google Scholar

[22]

D. J. B. Lloyd, B. Sandstede, D. Avitabile and A. R. Champneys, Localized hexagon patterns of the planar Swift-Hohenberg equation,, SIAM J. Appl. Dyn. Sys., 7 (2008), 1049. doi: 10.1137/070707622. Google Scholar

[23]

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis and J. Cuevas, Travelling solitary waves in the discrete Schrödinger equation with saturable nonlinearity: Existence, stability and dynamics,, Physica D, 237 (2008), 551. doi: 10.1016/j.physd.2007.09.026. Google Scholar

[24]

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis and J. Cuevas, Radiationless traveling waves in saturable nonlinear Schrödinger lattices,, Phys. Rev. Lett., 97 (2006). doi: 10.1103/PhysRevLett.97.124101. Google Scholar

[25]

T. R. O. Melvin, A. R. Champneys and D. E. Pelinovsky, Discrete traveling solitons in the Salerno model,, SIAM J. Appl. Dyn. Sys., 8 (2009), 689. doi: 10.1137/080715408. Google Scholar

[26]

O. F. Oxtoby and I. V. Barashenkov, Moving solitons in the discrete nonlinear Schrödinger equation,, Phys. Rev. E, 76 (2007). doi: 10.1103/PhysRevE.76.036603. Google Scholar

[27]

F. Pedaci, S. Barland, E. Caboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, G.-L. Oppo, G. Tissoni and R. Jäger, All-optical delay line using semiconductor cavity solitons,, Appl. Phys. Lett., 92 (2008). doi: 10.1063/1.2828458. Google Scholar

[28]

D. E. Pelinovsky, T. R. O. Melvin and A. R. Champneys, One-parameter localized traveling waves in nonlinear Schrödinger lattices,, Physica D, 236 (2007), 22. doi: 10.1016/j.physd.2007.07.010. Google Scholar

[29]

U. Peschel, O. A. Egorov and F. Lederer, Discrete cavity solitons,, Optics Letters, 29 (2004), 1909. doi: 10.1364/OL.29.001909. Google Scholar

[30]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics,, Physica D, 23 (1986), 3. doi: 10.1016/0167-2789(86)90104-1. Google Scholar

[31]

M. V. Shaskov and D. V. Turaev, An existence theorem of smooth nonlocal center manifolds for systems close to a system with a homoclinic loop,, J. Nonl. Sci., 9 (1999), 525. doi: 10.1007/s003329900078. Google Scholar

[32]

U. Thiele and E. Knobloch, Driven drops on heterogeneous substrates: Onset of sliding motion,, Phys. Rev. Lett., 97 (2006). doi: 10.1103/PhysRevLett.97.204501. Google Scholar

[33]

P. D. Woods and A. R. Champneys, Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamilton-Hopf bifurcation,, Physica D, 129 (1999), 147. doi: 10.1016/S0167-2789(98)00309-1. Google Scholar

[34]

A. V. Yulin and A. R. Champneys, Discrete snaking: Multiple cavity solitons in saturable media,, SIAM J Appl. Dyn. Sys., 9 (2010), 391. Google Scholar

[35]

A. V. Yulin, A. R. Champneys and D. V. Skryabin, Discrete cavity solitons due to saturable nonlinearity,, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.011804. Google Scholar

[36]

A. V. Yulin, O. A. Egorov, F. Lederer and D. V. Skryabin, Dark polariton solitons in semiconductor microcavities,, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.061801. Google Scholar

show all references

References:
[1]

M. Beck, J. Knobloch, D. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders and isolas of localized patterns,, SIAM J. Math. Anal., 41 (2009), 936. doi: 10.1137/080713306. Google Scholar

[2]

J. Burke, S. M. Houghton and E. Knobloch, Swift-Hohenberg equation with broken reflection symmetry,, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.036202. Google Scholar

[3]

J. Burke and E. Knobloch, Multipulse states in the Swift-Hohenberg equation, Discrete Contin. Dyn. Syst. 2009,, in, (): 109. Google Scholar

[4]

J. Burke and E. Knobloch, Homoclinic snaking: Structure and stability,, Chaos, 17 (2007). doi: 10.1063/1.2746816. Google Scholar

[5]

J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift-Hohenberg equation,, Phys. Lett. A, 360 (2007), 681. doi: 10.1016/j.physleta.2006.08.072. Google Scholar

[6]

P. Coullet, C. Riera and C. Tresser, Stable static localized structures in one dimension,, Phys. Rev. Lett., 84 (2000), 3069. doi: 10.1103/PhysRevLett.84.3069. Google Scholar

[7]

G. Dangelmayr, J. Hettel and E. Knobloch, Parity-breaking bifurcation in inhomogeneous systems, Nonlinearity, 10 (1997), 1093. doi: 10.1088/0951-7715/10/5/006. Google Scholar

[8]

J. H. P. Dawes, Localized pattern formation with a large scale mode: Slanted snaking,, SIAM J. Appl. Dyn. Syst., 7 (2008), 186. doi: 10.1137/06067794X. Google Scholar

[9]

O. A. Egorov, F. Lederer and Y. S. Kivshar, How does an inclined holding beam affect discrete modulational instability and solitons in nonlinear cavities?,, Optics Express, 15 (2007), 4149. doi: 10.1364/OE.15.004149. Google Scholar

[10]

O. Egorov, U. Peschel and F. Lederer, Mobility of discrete cavity solitons,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.066603. Google Scholar

[11]

O. A. Egorov, U. Peschel and F. Lederer, Discrete quadratic cavity solitons,, Phys. Rev. E, 71 (2005). doi: 10.1103/PhysRevE.71.056612. Google Scholar

[12]

O. A. Egorov, D. V. Skryabin, A. V. Yulin and F. Lederer, Bright cavity polariton solitons,, Phys. Rev. Lett., 102 (2009). doi: 10.1103/PhysRevLett.102.153904. Google Scholar

[13]

W. J. Firth and A. J. Scroggie, Optical bullet holes: Robust controllable localized states of a nonlinear cavity,, Phys. Rev. Lett., 76 (1996), 1623. doi: 10.1103/PhysRevLett.76.1623. Google Scholar

[14]

D. Gomilla and G. L. Oppo, Subcritical patterns and dissipative solitons due to intracavity photonic crystals,, Phys. Rev. A, 76 (2007). doi: 10.1103/PhysRevA.76.043823. Google Scholar

[15]

A. Hagberg and E. Meron, Propgation failure in excitable media,, Phys. Rev. E, 57 (1998), 229. doi: 10.1103/PhysRevE.57.299. Google Scholar

[16]

F. Haudin, R. G. Elias, R. G. Rojas, U. Bortolozzo, M. G. Clerc and S. Residori, Driven front propogation in 1D spatially periodic media,, Phys. Rev. Lett., 103 (2009). doi: 10.1103/PhysRevLett.103.128003. Google Scholar

[17]

P. G. Kevrekidis, "The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives,", Springer, (2009). doi: 10.1007/978-3-540-89199-4. Google Scholar

[18]

P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop and E. S. Titi, Continuum approach to discreteness,, Physical Review E, 65 (2002). doi: 10.1103/PhysRevE.65.046613. Google Scholar

[19]

J. Knobloch, D. Lloyd, B. Sandstede and T. Wagenknecht, Isolas of two-pulse solutions in homoclinic snaking scenarios,, preprint (2009)., (2009). Google Scholar

[20]

G. Kozyreff and S. J. Chapman, Asymptotics of large bound states of localized structures,, Phys. Rev. Lett., 97 (2006). doi: 10.1103/PhysRevLett.97.044502. Google Scholar

[21]

J. S. W. Lamb and H. W. Capel, Local bifurcations on the plane with reversing point group symmetry,, Chaos, 5 (1995), 271. doi: 10.1016/0960-0779(93)E0022-4. Google Scholar

[22]

D. J. B. Lloyd, B. Sandstede, D. Avitabile and A. R. Champneys, Localized hexagon patterns of the planar Swift-Hohenberg equation,, SIAM J. Appl. Dyn. Sys., 7 (2008), 1049. doi: 10.1137/070707622. Google Scholar

[23]

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis and J. Cuevas, Travelling solitary waves in the discrete Schrödinger equation with saturable nonlinearity: Existence, stability and dynamics,, Physica D, 237 (2008), 551. doi: 10.1016/j.physd.2007.09.026. Google Scholar

[24]

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis and J. Cuevas, Radiationless traveling waves in saturable nonlinear Schrödinger lattices,, Phys. Rev. Lett., 97 (2006). doi: 10.1103/PhysRevLett.97.124101. Google Scholar

[25]

T. R. O. Melvin, A. R. Champneys and D. E. Pelinovsky, Discrete traveling solitons in the Salerno model,, SIAM J. Appl. Dyn. Sys., 8 (2009), 689. doi: 10.1137/080715408. Google Scholar

[26]

O. F. Oxtoby and I. V. Barashenkov, Moving solitons in the discrete nonlinear Schrödinger equation,, Phys. Rev. E, 76 (2007). doi: 10.1103/PhysRevE.76.036603. Google Scholar

[27]

F. Pedaci, S. Barland, E. Caboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, G.-L. Oppo, G. Tissoni and R. Jäger, All-optical delay line using semiconductor cavity solitons,, Appl. Phys. Lett., 92 (2008). doi: 10.1063/1.2828458. Google Scholar

[28]

D. E. Pelinovsky, T. R. O. Melvin and A. R. Champneys, One-parameter localized traveling waves in nonlinear Schrödinger lattices,, Physica D, 236 (2007), 22. doi: 10.1016/j.physd.2007.07.010. Google Scholar

[29]

U. Peschel, O. A. Egorov and F. Lederer, Discrete cavity solitons,, Optics Letters, 29 (2004), 1909. doi: 10.1364/OL.29.001909. Google Scholar

[30]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics,, Physica D, 23 (1986), 3. doi: 10.1016/0167-2789(86)90104-1. Google Scholar

[31]

M. V. Shaskov and D. V. Turaev, An existence theorem of smooth nonlocal center manifolds for systems close to a system with a homoclinic loop,, J. Nonl. Sci., 9 (1999), 525. doi: 10.1007/s003329900078. Google Scholar

[32]

U. Thiele and E. Knobloch, Driven drops on heterogeneous substrates: Onset of sliding motion,, Phys. Rev. Lett., 97 (2006). doi: 10.1103/PhysRevLett.97.204501. Google Scholar

[33]

P. D. Woods and A. R. Champneys, Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamilton-Hopf bifurcation,, Physica D, 129 (1999), 147. doi: 10.1016/S0167-2789(98)00309-1. Google Scholar

[34]

A. V. Yulin and A. R. Champneys, Discrete snaking: Multiple cavity solitons in saturable media,, SIAM J Appl. Dyn. Sys., 9 (2010), 391. Google Scholar

[35]

A. V. Yulin, A. R. Champneys and D. V. Skryabin, Discrete cavity solitons due to saturable nonlinearity,, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.011804. Google Scholar

[36]

A. V. Yulin, O. A. Egorov, F. Lederer and D. V. Skryabin, Dark polariton solitons in semiconductor microcavities,, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.061801. Google Scholar

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