A singular perturbation problem arising from the Kuramoto-Sivashinsky equation

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.