I. Baldoma, M. Guardia, and D.E. Pelinovsky
On a countable sequence of homoclinic orbits arising near a saddle-center point
Abstract:
Exponential small splitting of separatrices in the singular perturbation
theory leads generally to nonvanishing oscillations near a center-saddle 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 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. The main technical difficulty in
the proof is to obtain estimates of the exponentially small terms in the complex plane
between the two symmetric pairs of singularities.
Keywords:
Traveling waves, beyond all orders, exponentially small splitting, center-saddle point, analytic continuation.