The use of Borel-summation in the establishment of non-existence of certain travelling-wave solutions of the Kuramoto-Sivashinsky equation
A method to prove the nonexistence of regular shocks for the Kuramoto- Sivashinsky equation \(\varepsilon^ 2u_{qqq} +(1-4 \varepsilon^ 2)u_ q=1-u^ 2\), \(u=u(q)\), when \(\varepsilon\) is sufficiently small, is developed. In a joint paper with \textit{A. P. Hooper} [Physica D 50, 231-238 (1991)] the author shows that instead of the condition \(u_{qq}=0\) at \(q=0\) we have \(u_{qq} (0) \sim -2 \gamma \varepsilon^{-4} \exp (-\pi (2 \varepsilon)^{-1})\), \(\varepsilon\to 0\). In this note an alternative procedure for finding \(\gamma\) is described. Using the Borel summation method the value of \(\gamma\) is calculated: \(\gamma=1,9652\dots\).
scientific article; zbMATH DE number 147211