Whitham's modulation equations and stability of periodic wave solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky Equation

Abstract

We study the spectral stability of periodic wave trains of the Korteweg-de Vries-Kuramoto-Sivashinsky equation which are, among many other applications, often used to describethe evo- lution of a thin liquid film flowing down an inclined ramp. More precisely, we show that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to side-band perturbations. Here, we use a direct Bloch ex- pansion method and spectral perturbation analysis instead of Evans function computations. We first establish, in our context, the now usual connection between first-order expansion of eigenvalues bi- furcating from the origin (both eigenvalue 0 and Floquet parameter 0) and the first-order Whitham's modulation system: the hyper- bolicity of such a system provides a necessary condition of spectral stability. Under a condition of strict hyperbolicity, we show that eigenvalues are indeed analytic in the neighborhood of the origin and that their expansion up to second order is connected to a vis- cous correction of the Whitham's equations. This, in turn, pro- vides new stability criteria. Finally, we study the Korteweg-de Vries limit: in this case, the domain of validity of the previous expansion shrinks to nothing and a new modulation theory is needed. The new modulation system consists of the Korteweg-de Vries modula- tion equations supplemented with a source term: relaxation limit in such a system provides, in turn, some stability criteria.