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Dark solitary waves in nonlocal nonlinear Schrödinger systems
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 |
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. |
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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