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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

Alexey Yulin 1, and  Alan Champneys 1,

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.
Keywords: solitary waves, homoclinic snaking., Lattice equation.
Mathematics Subject Classification: Primary: 37C29, 37L60; Secondary: 37K40, 37G2.
Citation: Alexey Yulin, Alan Champneys. Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities. Discrete and 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), 936-972. doi: 10.1137/080713306.

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J. Burke and E. Knobloch, Homoclinic snaking: Structure and stability, Chaos, 17 (2007), 037102. doi: 10.1063/1.2746816.

[5]

J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift-Hohenberg equation, Phys. Lett. A, 360 (2007), 681-688. doi: 10.1016/j.physleta.2006.08.072.

[6]

P. Coullet, C. Riera and C. Tresser, Stable static localized structures in one dimension, Phys. Rev. Lett., 84 (2000), 3069-3072. doi: 10.1103/PhysRevLett.84.3069.

[7]

G. Dangelmayr, J. Hettel and E. Knobloch, Parity-breaking bifurcation in inhomogeneous systems Nonlinearity, 10 (1997), 1093-1114. doi: 10.1088/0951-7715/10/5/006.

[8]

J. H. P. Dawes, Localized pattern formation with a large scale mode: Slanted snaking, SIAM J. Appl. Dyn. Syst., 7 (2008), 186-206. doi: 10.1137/06067794X.

[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), 4149-4158. doi: 10.1364/OE.15.004149.

[10]

O. Egorov, U. Peschel and F. Lederer, Mobility of discrete cavity solitons, Phys. Rev. E, 72 (2005), 066603. doi: 10.1103/PhysRevE.72.066603.

[11]

O. A. Egorov, U. Peschel and F. Lederer, Discrete quadratic cavity solitons, Phys. Rev. E, 71 (2005), 056612. doi: 10.1103/PhysRevE.71.056612.

[12]

O. A. Egorov, D. V. Skryabin, A. V. Yulin and F. Lederer, Bright cavity polariton solitons, Phys. Rev. Lett., 102 (2009), 153904. doi: 10.1103/PhysRevLett.102.153904.

[13]

W. J. Firth and A. J. Scroggie, Optical bullet holes: Robust controllable localized states of a nonlinear cavity, Phys. Rev. Lett., 76 (1996), 1623-1626. doi: 10.1103/PhysRevLett.76.1623.

[14]

D. Gomilla and G. L. Oppo, Subcritical patterns and dissipative solitons due to intracavity photonic crystals, Phys. Rev. A, 76 (2007), 043823. doi: 10.1103/PhysRevA.76.043823.

[15]

A. Hagberg and E. Meron, Propgation failure in excitable media, Phys. Rev. E, 57 (1998), 229-303. doi: 10.1103/PhysRevE.57.299.

[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), 128003. doi: 10.1103/PhysRevLett.103.128003.

[17]

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

[18]

P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop and E. S. Titi, Continuum approach to discreteness, Physical Review E, 65 (2002), 046613. doi: 10.1103/PhysRevE.65.046613.

[19]

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

[20]

G. Kozyreff and S. J. Chapman, Asymptotics of large bound states of localized structures, Phys. Rev. Lett., 97 (2006), 044502. doi: 10.1103/PhysRevLett.97.044502.

[21]

J. S. W. Lamb and H. W. Capel, Local bifurcations on the plane with reversing point group symmetry, Chaos, Solitons and Fractals, 5 (1995), 271-293. doi: 10.1016/0960-0779(93)E0022-4.

[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), 1049-1100. doi: 10.1137/070707622.

[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), 551-567. doi: 10.1016/j.physd.2007.09.026.

[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), 124101. doi: 10.1103/PhysRevLett.97.124101.

[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), 689-709. doi: 10.1137/080715408.

[26]

O. F. Oxtoby and I. V. Barashenkov, Moving solitons in the discrete nonlinear Schrödinger equation, Phys. Rev. E, 76 (2007), 036603. doi: 10.1103/PhysRevE.76.036603.

[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), 011101. doi: 10.1063/1.2828458.

[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), 22-43. doi: 10.1016/j.physd.2007.07.010.

[29]

U. Peschel, O. A. Egorov and F. Lederer, Discrete cavity solitons, Optics Letters, 29 (2004), 1909-1911. doi: 10.1364/OL.29.001909.

[30]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D, 23 (1986), 3-11. doi: 10.1016/0167-2789(86)90104-1.

[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), 525-573. doi: 10.1007/s003329900078.

[32]

U. Thiele and E. Knobloch, Driven drops on heterogeneous substrates: Onset of sliding motion, Phys. Rev. Lett., 97 (2006), 204501. doi: 10.1103/PhysRevLett.97.204501.

[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), 147-170. doi: 10.1016/S0167-2789(98)00309-1.

[34]

A. V. Yulin and A. R. Champneys, Discrete snaking: Multiple cavity solitons in saturable media, SIAM J Appl. Dyn. Sys., 9 (2010), 391-431.

[35]

A. V. Yulin, A. R. Champneys and D. V. Skryabin, Discrete cavity solitons due to saturable nonlinearity, Phys. Rev. A, 78 (2008), 011804(R). doi: 10.1103/PhysRevA.78.011804.

[36]

A. V. Yulin, O. A. Egorov, F. Lederer and D. V. Skryabin, Dark polariton solitons in semiconductor microcavities, Phys. Rev. A, 78 (2008), 061801. doi: 10.1103/PhysRevA.78.061801.

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), 936-972. doi: 10.1137/080713306.

[2]

J. Burke, S. M. Houghton and E. Knobloch, Swift-Hohenberg equation with broken reflection symmetry, Phys. Rev. E, 80 (2009), 036202. doi: 10.1103/PhysRevE.80.036202.

[3]

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

[4]

J. Burke and E. Knobloch, Homoclinic snaking: Structure and stability, Chaos, 17 (2007), 037102. doi: 10.1063/1.2746816.

[5]

J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift-Hohenberg equation, Phys. Lett. A, 360 (2007), 681-688. doi: 10.1016/j.physleta.2006.08.072.

[6]

P. Coullet, C. Riera and C. Tresser, Stable static localized structures in one dimension, Phys. Rev. Lett., 84 (2000), 3069-3072. doi: 10.1103/PhysRevLett.84.3069.

[7]

G. Dangelmayr, J. Hettel and E. Knobloch, Parity-breaking bifurcation in inhomogeneous systems Nonlinearity, 10 (1997), 1093-1114. doi: 10.1088/0951-7715/10/5/006.

[8]

J. H. P. Dawes, Localized pattern formation with a large scale mode: Slanted snaking, SIAM J. Appl. Dyn. Syst., 7 (2008), 186-206. doi: 10.1137/06067794X.

[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), 4149-4158. doi: 10.1364/OE.15.004149.

[10]

O. Egorov, U. Peschel and F. Lederer, Mobility of discrete cavity solitons, Phys. Rev. E, 72 (2005), 066603. doi: 10.1103/PhysRevE.72.066603.

[11]

O. A. Egorov, U. Peschel and F. Lederer, Discrete quadratic cavity solitons, Phys. Rev. E, 71 (2005), 056612. doi: 10.1103/PhysRevE.71.056612.

[12]

O. A. Egorov, D. V. Skryabin, A. V. Yulin and F. Lederer, Bright cavity polariton solitons, Phys. Rev. Lett., 102 (2009), 153904. doi: 10.1103/PhysRevLett.102.153904.

[13]

W. J. Firth and A. J. Scroggie, Optical bullet holes: Robust controllable localized states of a nonlinear cavity, Phys. Rev. Lett., 76 (1996), 1623-1626. doi: 10.1103/PhysRevLett.76.1623.

[14]

D. Gomilla and G. L. Oppo, Subcritical patterns and dissipative solitons due to intracavity photonic crystals, Phys. Rev. A, 76 (2007), 043823. doi: 10.1103/PhysRevA.76.043823.

[15]

A. Hagberg and E. Meron, Propgation failure in excitable media, Phys. Rev. E, 57 (1998), 229-303. doi: 10.1103/PhysRevE.57.299.

[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), 128003. doi: 10.1103/PhysRevLett.103.128003.

[17]

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

[18]

P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop and E. S. Titi, Continuum approach to discreteness, Physical Review E, 65 (2002), 046613. doi: 10.1103/PhysRevE.65.046613.

[19]

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

[20]

G. Kozyreff and S. J. Chapman, Asymptotics of large bound states of localized structures, Phys. Rev. Lett., 97 (2006), 044502. doi: 10.1103/PhysRevLett.97.044502.

[21]

J. S. W. Lamb and H. W. Capel, Local bifurcations on the plane with reversing point group symmetry, Chaos, Solitons and Fractals, 5 (1995), 271-293. doi: 10.1016/0960-0779(93)E0022-4.

[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), 1049-1100. doi: 10.1137/070707622.

[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), 551-567. doi: 10.1016/j.physd.2007.09.026.

[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), 124101. doi: 10.1103/PhysRevLett.97.124101.

[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), 689-709. doi: 10.1137/080715408.

[26]

O. F. Oxtoby and I. V. Barashenkov, Moving solitons in the discrete nonlinear Schrödinger equation, Phys. Rev. E, 76 (2007), 036603. doi: 10.1103/PhysRevE.76.036603.

[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), 011101. doi: 10.1063/1.2828458.

[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), 22-43. doi: 10.1016/j.physd.2007.07.010.

[29]

U. Peschel, O. A. Egorov and F. Lederer, Discrete cavity solitons, Optics Letters, 29 (2004), 1909-1911. doi: 10.1364/OL.29.001909.

[30]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D, 23 (1986), 3-11. doi: 10.1016/0167-2789(86)90104-1.

[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), 525-573. doi: 10.1007/s003329900078.

[32]

U. Thiele and E. Knobloch, Driven drops on heterogeneous substrates: Onset of sliding motion, Phys. Rev. Lett., 97 (2006), 204501. doi: 10.1103/PhysRevLett.97.204501.

[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), 147-170. doi: 10.1016/S0167-2789(98)00309-1.

[34]

A. V. Yulin and A. R. Champneys, Discrete snaking: Multiple cavity solitons in saturable media, SIAM J Appl. Dyn. Sys., 9 (2010), 391-431.

[35]

A. V. Yulin, A. R. Champneys and D. V. Skryabin, Discrete cavity solitons due to saturable nonlinearity, Phys. Rev. A, 78 (2008), 011804(R). doi: 10.1103/PhysRevA.78.011804.

[36]

A. V. Yulin, O. A. Egorov, F. Lederer and D. V. Skryabin, Dark polariton solitons in semiconductor microcavities, Phys. Rev. A, 78 (2008), 061801. doi: 10.1103/PhysRevA.78.061801.

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