On a countable sequence of homoclinic orbits arising near a saddle-center point

TL;DR

This paper proves a long-standing conjecture that oscillations near a saddle-center can vanish at a countable set of parameters due to complex singularities, demonstrated through a specific fourth-order equation related to the mKdV equation.

Contribution

It provides a rigorous proof of the conjecture linking complex singularities to the vanishing of oscillations in homoclinic orbits for a particular nonlinear PDE.

Findings

Confirmed the conjecture for a specific fourth-order equation.

Established the connection between complex singularities and homoclinic orbit behavior.

Demonstrated the existence of a countable set of parameter values with vanishing oscillations.

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.