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

    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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