Title

Breaking homoclinic connections for a singularly perturbed differential equation and the Stokes phenomenon

Authors

Authors

A. Tovbis

Comments

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Abbreviated Journal Title

Stud. Appl. Math.

Keywords

ASYMPTOTICS; ORDERS; Mathematics, Applied

Abstract

Behavior of the separatrix solution y(t) = -(3/2)/cosh(2) (t/2) (homoclinic connection) of the second order equation y" = y + y(2) that undergoes the singular perturbation epsilon (2)y'''' + y" = y + y(2), where epsilon > 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 epsilon -- > 0) regular singular points, where the Stokes phenomenon plays the role of the leading-order approximation.

Journal Title

Studies in Applied Mathematics

Volume

104

Issue/Number

4

Publication Date

1-1-2000

Document Type

Article

Language

English

First Page

353

Last Page

386

WOS Identifier

WOS:000090135500003

ISSN

0022-2526

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