Whitham's Modulation Equations and Stability of Periodic Wave Solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky Equation

Pascal Noble (1) , Luis Miguel Miguel Rodrigues (1)

1 ICJ - Institut Camille Jordan

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 describe the evolution 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 expansion 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 bifurcating from the origin (both eigenvalue 0 and Floquet parameter 0) and the first order Whitham's modulation system: the hyperbolicity 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 viscous correction of the Whitham's equations. This, in turn, provides 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 in the Korteweg-de Vries modulation equations supplemented with a source term: relaxation limit in such a system provides in turn some stability criteria.

Domaines: Equations aux dérivées partielles [math.AP]

Pascal Noble, Luis Miguel Miguel Rodrigues. Whitham's Modulation Equations and Stability of Periodic Wave Solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky Equation. Indiana University Mathematics Journal, 2013, 62 (3), pp.753-783. ⟨hal-00943713⟩