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2006.389: When gap solitons become embedded solitons: a generic unfolding

2006.389: T. Wagenknecht and A. R. Champneys (2003) When gap solitons become embedded solitons: a generic unfolding. Physica D, 177. pp. 50-70. ISSN 0167-2789

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DOI: 10.1016/S0167-2789(02)00773-X


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
Uncontrolled Keywords:Embedded soliton; Homoclinic bifurcation; Degenerate equilibrium; Reversible system
Subjects:PACS 2003 > 05 Statistical physics, thermodynamics, and nonlinear dynamical systems
PACS 2003 > 42 Optics
MIMS number:2006.389
Deposited By:Thomas Wagenknecht
Deposited On:17 October 2006

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