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A. Tovbis and D. Pelinovsky

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Exact conditions for existence of homoclinic orbits in the fifth-order KdV model

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Nonlinearity 19, 2277-2312 (2006)

**Abstract:**

We consider homoclinic orbits in the fourth-order differential equation that
occurs in a reduction of the fifth-order KdV model. Numerous computations show that
homoclinic orbits exist on certain curves in the two-parameter
plane. We study the curves in the beyond-all-order limit and prove
that a curve passes through the point only if the Stokes constant for the
truncated equation vanishes. Additional condition that the derivative of the Stokes constant
does not vanish guarantees the existence of a unique curve passing
through the point. Every homoclinic orbit is proved to be
single-humped sufficiently close to the beyond-all-order limit.

**Keywords**:

FIFTH-ORDER KDV EQUATION, HOMOCLINIC ORBITS, BEYOND-ALL-ORDERS ASYMPTOTIC EXPANSIONS,
INNER AND OUTER PERTURBATION SERIES, STOKES CONSTANTS