Title

Breaking Homoclinic Connections For A Singularly Perturbed Differential Equation And The Stokes Phenomenon

Abstract

Behavior of the separatrix solution y(t) = -(3/2)/ cosh2(t/2) (homoclinic connection) of the second order equation y″ = y + y2 that undergoes the singular perturbation ε2y″″ + y″ = y + y2, where ε > 0 is a small parameter, is considered. This equation arises in the theory of traveling water waves in the presence of surface tension. It has been demonstrated both rigorously [1,2] and using formal asymptotic arguments [3,4] that the above-mentioned solution could not survive the perturbation. The latter papers were based on the Kruskal-Segur method (KS method), originally developed for the equation of crystal growth [5]. In fact, the key point of this method is the reduction of the original problem to the Stokes phenomenon of a certain parameterless "leading-order" equation. The main purpose of this article is further development of the KS method to study the breaking of homoclinic connections. In particular: (1) a rigorous basis for the KS method in the case of the above-mentioned perturbed problem is provided; and (2) it is demonstrated that breaking of a homoclinic connection is reducible to a monodromy problem for coalescing (as ε → 0) regular singular points, where the Stokes phenomenon plays the role of the leading-order approximation.

Publication Date

1-1-2000

Publication Title

Studies in Applied Mathematics

Volume

104

Issue

4

Number of Pages

353-386

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1111/1467-9590.00138

Socpus ID

0039621056 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/0039621056

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