Wagenknecht, T. and Champneys, A. R.
(2003)
*When gap solitons become embedded solitons: a generic unfolding.*
Physica D, 177.
pp. 50-70.
ISSN 0167-2789

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## Abstract

A two-parameter unfolding is considered of single-pulsed homoclinic orbits to an equilibrium with two real and two zero eigenvalues in fourth-order reversible dynamical systems. One parameter controls the linearisation, with a transition occurring between a saddle-centre and a hyperbolic equilibrium. In the saddle-centre region, the homoclinic orbit is of codimension-one, which is controlled by the second generic parameter, whereas when the equilibrium is hyperbolic the homoclinic orbit is structurally stable. A geometric approach reveals the homoclinic orbits to the saddle to be generically destroyed either by developing an algebraically decaying tail or through a fold, depending on the sign of the perturbation of the second parameter. Special cases of different actions of Z2-symmetry are considered, as is the case of the system being Hamiltonian. Application of these results is considered to the transition between embedded solitons (corresponding to the codimension-one-homoclinic orbits) and gap solitons (the structurally stable ones) in nonlinear wave systems. The theory is shown to match numerical experiments on two models arising in nonlinear optics and on a form of fifth-order Korteweg de Vries equation.

Item Type: | Article |
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Uncontrolled Keywords: | Embedded soliton; Homoclinic bifurcation; Degenerate equilibrium; Reversible system |

Subjects: | PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 00 GENERAL PHYSICS > 05 Statistical physics, thermodynamics, and nonlinear dynamical systems PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 40 ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS > 42 Optics |

Depositing User: | Thomas Wagenknecht |

Date Deposited: | 17 Oct 2006 |

Last Modified: | 20 Oct 2017 14:12 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/628 |

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