Beyond-all-orders effects in multiple-scales asymptotics: travelling-wave solutions to the Kuramoto-Sivashinsky equation

This paper concerns the possible 'shock' patterns that can exist in the solution to a singularly perturbed, third-order nonlinear ordinary differential equation arising as the travelling-wave reduction of the Kuramoto-Sivashinsky equation. In particular, the existence (or otherwise) of osc...

Full description

Saved in:
Bibliographic Details
Published in:Journal of engineering mathematics 2003-04, Vol.45 (3-4), p.197-226
Main Authors: Adams, K L, King, J R, Tew, R H
Format: Article
Language:English
Citations: Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:This paper concerns the possible 'shock' patterns that can exist in the solution to a singularly perturbed, third-order nonlinear ordinary differential equation arising as the travelling-wave reduction of the Kuramoto-Sivashinsky equation. In particular, the existence (or otherwise) of oscillatory shocks and multiple shocks made up of combinations of oscillatory and monotonic shocks is examined, using an optimal truncation strategy to track crucial exponentially small terms lying beyond all orders of the (divergent) algebraic expansion. The results provide further understanding of numerical solutions previously obtained by others, as well as giving a methodology which is applicable to much broader classes of differential equations exhibiting multiscale phenomena; they also afford same new insight into the multi-scales technique.
ISSN:0022-0833
DOI:10.1023/A:1022600411856