Beyond-all-orders effects in multiple-scales asymptotics: travelling-wave solutions to the Kuramoto-Sivashinsky equation

This paper concerns the possible 'shock' patterns that can exist in the solution to a singularly perturbed, third-order nonlinear ordinary differential equation arising as the travelling-wave reduction of the Kuramoto-Sivashinsky equation. In particular, the existence (or otherwise) of oscillatory shocks and multiple shocks made up of combinations of oscillatory and monotonic shocks is examined, using an optimal truncation strategy to track crucial exponentially small terms lying beyond all orders of the (divergent) algebraic expansion. The results provide further understanding of numerical solutions previously obtained by others, as well as giving a methodology which is applicable to much broader classes of differential equations exhibiting multiscale phenomena; they also afford same new insight into the multi-scales technique.