CaltechAUTHORS
  A Caltech Library Service

Exponentially small estimates for separatrix splittings

Scheurle, Jürgen and Marsden, Jerrold E. and Holmes, Philip (1991) Exponentially small estimates for separatrix splittings. In: Asymptotics beyond All Orders. NATO Science Series B: Physics. No.284. Plenum , pp. 187-195. ISBN 9780306441127 http://resolver.caltech.edu/CaltechAUTHORS:20100917-095953372

[img]
Preview
PDF - Updated Version
See Usage Policy.

112Kb

Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20100917-095953372

Abstract

This paper reviews our previous estimates and gives an example exhibiting a new phenomenon. In problems involving asymptotics beyond all orders in a perturbation parameter є, it is a common assumption that the quantity being studied (such as a separatrix splitting distance or angle, a solitary wave mismatch, etc.) can be “estimated” by an expression of the form aє^be^(−c/є) as є → 0. Here, a, b and c are constants (where b can be negative and c is “sharp”, often the distance from the real axis to a pole in the complex plane). The main purpose of our example is to show that this assumption can be wrong. The example, which concerns the splitting of separatrices in a rapidly forced system with a heteroclinic orbit shows that even the estimate from above (using the sharp value of c) can be incorrect. We argue that this situation is not isolated or particular, but happens rather generally. We especially note that in situations involving asymptotics beyond all orders, when an estimate of the form aє^be^(−c/є) is assumed, it needs to be justified.


Item Type:Book Section
Additional Information:© 1991, Plenum Press. January, 1991, this version, June, 91. Research partially supported by NSF grant DMS 89-22704 and a Humboldt award during a visit to the Universität Hamburg. We thank Martin Kummer, Jim Ellison, Harvey Segur, and Saleh Tanveer for several helpful suggestions.
Funders:
Funding AgencyGrant Number
NSFDMS 89-22704
Record Number:CaltechAUTHORS:20100917-095953372
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100917-095953372
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20010
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:17 Sep 2010 21:06
Last Modified:26 Dec 2012 12:26

Repository Staff Only: item control page