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1997 A singular perturbation problem arising from the Kuramoto-Sivashinsky equation
J. B. McLeod, S. V. Raghavan, W. C. Troy
Differential Integral Equations 10(1): 1-36 (1997).

Abstract

In looking for particular solutions to the Kuramoto-Sivashinsky equation the following ordinary differential equation arises: $$ \epsilon w^{\prime\prime\prime} +w^\prime = 1- w^2. $$ In some applications $\epsilon$ is small and a singular perturbation problem arises. In this paper we first show that for each $\epsilon>0$, there exists a unique monotonic solution which satisfies $w(0)=0 $ and $w(\infty)=1$. We also show that this solution is not odd by demonstrating that $w^{\prime\prime}(0) \ne 0$. Finally, we prove that $w^{\prime\prime}0)$ is asymptotically small beyond all orders of $\epsilon$ and we derive the asymptotic formula $w^{\prime\prime}(0) \sim -A \epsilon^{-2} \hbox{exp}\left(\frac{-\pi}{2 \sqrt \epsilon } \right) \rm{as}\, \epsilon \to 0$, where $A$ is some positive constant.

Citation

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J. B. McLeod. S. V. Raghavan. W. C. Troy. "A singular perturbation problem arising from the Kuramoto-Sivashinsky equation." Differential Integral Equations 10 (1) 1 - 36, 1997.

Information

Published: 1997
First available in Project Euclid: 6 May 2013

zbMATH: 0879.34028
MathSciNet: MR1424796

Subjects:
Primary: 34E15
Secondary: 34C37, 35Q53

Rights: Copyright © 1997 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.10 • No. 1 • 1997
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