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When gap solitons become embedded solitons : an unfolding via Lin's method

Research output: Working paperWorking paper and Preprints

Original languageEnglish
Publication date2002


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. Using an adaptation of Lin's method to deal with the non-hyperbolic equilibrium, the homoclinic orbits to the saddle are shown generically to be 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 $\Bbb{Z}_2$-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 5th-order Korteweg de Vries equation.

Additional information

Additional information: Preprint of a paper later published by Elsevier Science (2003), Physica D : Nonlinear phenomena, 177 (1-4), pp.50-70, ISSN 0167-2789



  • Bcanm 2002r04

    Preprint (usually an early version) , 426 KB, PDF-document

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