When gap solitons become embedded solitons: a generic unfolding

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
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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