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<p>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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月1日

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  • Inmaculada Baldomá,
  • Marcel Guardia,
  • Dmitry E. Pelinovsky

摘要

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.