Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities
Pages: 1341 - 1357, Volume 4, Issue 5, October 2011

doi:10.3934/dcdss.2011.4.1341      Abstract        References        Full text (1134.1K)           Related Articles

Alexey Yulin - Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, United Kingdom (email)
Alan Champneys - Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, United Kingdom (email)

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.
2	J. Burke, S. M. Houghton and E. Knobloch, Swift-Hohenberg equation with broken reflection symmetry, Phys. Rev. E, 80 (2009), 036202.
3	J. Burke and E. Knobloch, Multipulse states in the Swift-Hohenberg equation, Discrete Contin. Dyn. Syst. 2009, in "Dynamical Systems, Differential Equations and Applications," 7th AIMS Conference, suppl., 109-117.
4	J. Burke and E. Knobloch, Homoclinic snaking: Structure and stability, Chaos, 17 (2007), 037102.
5	J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift-Hohenberg equation, Phys. Lett. A, 360 (2007), 681-688.
6	P. Coullet, C. Riera and C. Tresser, Stable static localized structures in one dimension, Phys. Rev. Lett., 84 (2000), 3069-3072.
7	G. Dangelmayr, J. Hettel and E. Knobloch, Parity-breaking bifurcation in inhomogeneous systems Nonlinearity, 10 (1997), 1093-1114.
8	J. H. P. Dawes, Localized pattern formation with a large scale mode: Slanted snaking, SIAM J. Appl. Dyn. Syst., 7 (2008), 186-206.
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.
10	O. Egorov, U. Peschel and F. Lederer, Mobility of discrete cavity solitons, Phys. Rev. E, 72 (2005), 066603.
11	O. A. Egorov, U. Peschel and F. Lederer, Discrete quadratic cavity solitons, Phys. Rev. E, 71 (2005), 056612.
12	O. A. Egorov, D. V. Skryabin, A. V. Yulin and F. Lederer, Bright cavity polariton solitons, Phys. Rev. Lett., 102 (2009), 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.
14	D. Gomilla and G. L. Oppo, Subcritical patterns and dissipative solitons due to intracavity photonic crystals, Phys. Rev. A, 76 (2007), 043823.
15	A. Hagberg and E. Meron, Propgation failure in excitable media, Phys. Rev. E, 57 (1998), 229-303.
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.
17	P. G. Kevrekidis, "The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives," Springer, Berlin Heidelberg, 2009.
18	P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop and E. S. Titi, Continuum approach to discreteness, Physical Review E, 65 (2002), 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.
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.
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.
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.
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.
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.
26	O. F. Oxtoby and I. V. Barashenkov, Moving solitons in the discrete nonlinear Schrödinger equation, Phys. Rev. E, 76 (2007), 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.
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.
29	U. Peschel, O. A. Egorov and F. Lederer, Discrete cavity solitons, Optics Letters, 29 (2004), 1909-1911.
30	Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D, 23 (1986), 3-11.
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.
32	U. Thiele and E. Knobloch, Driven drops on heterogeneous substrates: Onset of sliding motion, Phys. Rev. Lett., 97 (2006), 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.
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).
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.

Go to top