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This paper concerns homoclinic solutions to periodic orbits in a fourth-order Hamiltonian system arising from a reduction of the classical water-wave problem in the presence of surface tension. These solutions correspond to travelling solitary waves which converge to non-decaying ripples at infinity. An analytical result of Amick and Toland, showing the existence of such homoclinic orbits to small amplitude periodic orbits in a singular limit, is extended numerically. Also, a related result by Amick and McLeod, showing the non-existence of homoclinic solutions to zero, is motivated geometrically. A general boundary-value method is constructed for continuation of homoclinic orbits to periodic orbits in Hamiltonian and reversible systems. Numerical results are presented using the path-following software AUTO, showing that the Amick-Toland solutions persist well away from the singular limit and for large-amplitude periodic orbits. Special account is taken of the phase shift between the two periodic solutions in the asymptotic limits. Furthermore, new multi-modal homoclinic solutions to periodic orbits are shown to exist under a transversality hypothesis, which is verified a posteriori by explicit computation. Continuation of these new solutions reveals limit points with respect to the singular parameter
Original languageEnglish
Publication statusPublished - Jun 1996

Bibliographical note

Additional information: Preprint of a paper later published by Elsevier Science, (1997) Physica D: Nonlinear Phenomena, 102 (1-2), pp. 101-124. ISSN 0167-2789

Sponsorship: GJ Lord acknowledges the support of the UK EPSRC

Structured keywords

  • Engineering Mathematics Research Group


  • homoclinic orbits
  • reversible systems
  • multi-modal homoclinic solutions
  • classical water-wave problem
  • periodic orbits
  • travelling solitary waves
  • continuation
  • fourth-order Hamiltonian systems

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