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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 & Continuous Dynamical Systems - S, 2011, 4 (5) : 1341-1357. doi: 10.3934/dcdss.2011.4.1341
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show all references

References:
[1]

SIAM J. Math. Anal., 41 (2009), 936-972. doi: 10.1137/080713306.  Google Scholar

[2]

Phys. Rev. E, 80 (2009), 036202. 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]

Chaos, 17 (2007), 037102. doi: 10.1063/1.2746816.  Google Scholar

[5]

Phys. Lett. A, 360 (2007), 681-688. doi: 10.1016/j.physleta.2006.08.072.  Google Scholar

[6]

Phys. Rev. Lett., 84 (2000), 3069-3072. doi: 10.1103/PhysRevLett.84.3069.  Google Scholar

[7]

Nonlinearity, 10 (1997), 1093-1114. doi: 10.1088/0951-7715/10/5/006.  Google Scholar

[8]

SIAM J. Appl. Dyn. Syst., 7 (2008), 186-206. doi: 10.1137/06067794X.  Google Scholar

[9]

Optics Express, 15 (2007), 4149-4158. doi: 10.1364/OE.15.004149.  Google Scholar

[10]

Phys. Rev. E, 72 (2005), 066603. doi: 10.1103/PhysRevE.72.066603.  Google Scholar

[11]

Phys. Rev. E, 71 (2005), 056612. doi: 10.1103/PhysRevE.71.056612.  Google Scholar

[12]

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[13]

Phys. Rev. Lett., 76 (1996), 1623-1626. doi: 10.1103/PhysRevLett.76.1623.  Google Scholar

[14]

Phys. Rev. A, 76 (2007), 043823. doi: 10.1103/PhysRevA.76.043823.  Google Scholar

[15]

Phys. Rev. E, 57 (1998), 229-303. doi: 10.1103/PhysRevE.57.299.  Google Scholar

[16]

Phys. Rev. Lett., 103 (2009), 128003. doi: 10.1103/PhysRevLett.103.128003.  Google Scholar

[17]

Springer, Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-89199-4.  Google Scholar

[18]

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[19]

preprint (2009). Google Scholar

[20]

Phys. Rev. Lett., 97 (2006), 044502. doi: 10.1103/PhysRevLett.97.044502.  Google Scholar

[21]

Chaos, Solitons and Fractals, 5 (1995), 271-293. doi: 10.1016/0960-0779(93)E0022-4.  Google Scholar

[22]

SIAM J. Appl. Dyn. Sys., 7 (2008), 1049-1100. doi: 10.1137/070707622.  Google Scholar

[23]

Physica D, 237 (2008), 551-567. doi: 10.1016/j.physd.2007.09.026.  Google Scholar

[24]

Phys. Rev. Lett., 97 (2006), 124101. doi: 10.1103/PhysRevLett.97.124101.  Google Scholar

[25]

SIAM J. Appl. Dyn. Sys., 8 (2009), 689-709. doi: 10.1137/080715408.  Google Scholar

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Phys. Rev. E, 76 (2007), 036603. doi: 10.1103/PhysRevE.76.036603.  Google Scholar

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[30]

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[31]

J. Nonl. Sci., 9 (1999), 525-573. doi: 10.1007/s003329900078.  Google Scholar

[32]

Phys. Rev. Lett., 97 (2006), 204501. doi: 10.1103/PhysRevLett.97.204501.  Google Scholar

[33]

Physica D, 129 (1999), 147-170. doi: 10.1016/S0167-2789(98)00309-1.  Google Scholar

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SIAM J Appl. Dyn. Sys., 9 (2010), 391-431.  Google Scholar

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Phys. Rev. A, 78 (2008), 011804(R). doi: 10.1103/PhysRevA.78.011804.  Google Scholar

[36]

Phys. Rev. A, 78 (2008), 061801. doi: 10.1103/PhysRevA.78.061801.  Google Scholar

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