Differential and Integral Equations

A singular perturbation problem arising from the Kuramoto-Sivashinsky equation

J. B. McLeod, S. V. Raghavan, and W. C. Troy

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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.

Article information

Differential Integral Equations, Volume 10, Number 1 (1997), 1-36.

First available in Project Euclid: 6 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34E15: Singular perturbations, general theory
Secondary: 34C37: Homoclinic and heteroclinic solutions 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]


Raghavan, S. V.; McLeod, J. B.; Troy, W. C. A singular perturbation problem arising from the Kuramoto-Sivashinsky equation. Differential Integral Equations 10 (1997), no. 1, 1--36. https://projecteuclid.org/euclid.die/1367846881

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