On a Countable Sequence of Homoclinic Orbits Arising Near a Saddle–Center Point

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Abstract
Exponential small splitting of separatrices in the singular perturbation theory leads generally to nonvanishing oscillations near a saddle–center point and to nonexistence of a true homoclinic orbit. It was conjectured long ago that the oscillations may vanish at a countable set of small parameter values if there exist a quadruplet of singularities in the complex analytic extension of the limiting homoclinic orbit. The present paper gives a rigorous proof of this conjecture for a particular fourth-order equation relevant to the traveling wave reduction of the modified Korteweg–de Vries equation with the fifth-order dispersion term.
Funding Information
  • Agencia Estatal de Investigación (PID-2021-122954NB-100, PID-2021-122954NB-100)
  • Institució Catalana de Recerca i Estudis Avançats (Not applicable)
  • H2020 European Research Council (757802, CEX2020-001084-M, CEX2020-001084-M)
  • Universitat de Barcelona

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