Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Table of Contents - 2007 - Volume 137, Issue 01  


Long-time persistence of Korteweg–de Vries solitons as transient dynamics in a model of inclined film flow

Robert L. Pego a1, Guido Schneider a2 and Hannes Uecker a2
a1 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (rpego@cmu.edu)
a2 Mathematisches Institut I, Universität Karlsruhe, 76128 Karlsruhe, Germany (guido.schneider@math.uni.karlsruhe.de; hannes.uecker@math.uni.karlsruhe.de)

Article author query
pego rl   [Google Scholar] 
schneider g   [Google Scholar] 
uecker h   [Google Scholar] 
 

Abstract

The Kuramoto–Sivashinsky-perturbed Korteweg–de Vries (KS–KdV) equation

$$ \partial_tu=-\partial_x^3u-\tfrac{1}{2}\partial_x(u^2)-\varepsilon(\partial_x^2+\partial_x^4)u, $$

with $0<\varepsilon\ll1$ a small parameter, arises as an amplitude equation for small amplitude long waves on the surface of a viscous liquid running down an inclined plane in certain regimes when the trivial solution, the so-called Nusselt solution, is sideband unstable. Although individual pulses are unstable due to the long-wave instability of the flat surface, the dynamics of KS–KdV is dominated by travelling pulse trains of $O(1)$ amplitude. As a step toward explaining the persistence of pulses and understanding their interactions, we prove that for $n=1$ and $2$ the KdV manifolds of $n$-solitons are stable in KS–KdV on an $O(1/\varepsilon)$ time-scale with respect to $O(1)$ perturbations in $H^n(\mathbb{R})$.

(Published Online February 9 2007)
(Received October 13 2005)
(Accepted January 25 2006)