Long-time persistence of Korteweg–de Vries solitons as transient dynamics in a model of inclined film flow

Abstract

The Kuramoto–Sivashinsky-perturbed Korteweg–de Vries (KS–KdV) equation$$ \partial_tu=-\partial_x^3u-\tfrac{1}{2}\partial_x(u^2)-\varepsilon(\partial_x^2+\partial_x^4)u, $$with $0<\varepsilon\ll1$ a small parameter, arises as an amplitude equation for small amplitude long waves on the surface of a viscous liquid running down an inclined plane in certain regimes when the trivial solution, the so-called Nusselt solution, is sideband unstable. Although individual pulses are unstable due to the long-wave instability of the flat surface, the dynamics of KS–KdV is dominated by travelling pulse trains of $O(1)$ amplitude. As a step toward explaining the persistence of pulses and understanding their interactions, we prove that for $n=1$ and $2$ the KdV manifolds of $n$-solitons are stable in KS–KdV on an $O(1/\varepsilon)$ time-scale with respect to $O(1)$ perturbations in $H^n(\mathbb{R})$.