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

## TL;DR

This paper proves a mathematical conjecture about oscillations near a specific point in a fourth-order equation related to wave dynamics.

## Contribution

The paper provides a rigorous proof of a long-standing conjecture regarding homoclinic orbits in singular perturbation theory.

## Key findings

- Oscillations near a saddle–center point vanish at a countable set of small parameter values.
- A quadruplet of singularities in the complex analytic extension leads to the vanishing oscillations.
- The proof applies to a fourth-order equation relevant to the modified Korteweg–de Vries equation.

## 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.