Whitham's modulation equations and stability of periodic wave solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky Equation Pascal NobleL. Miguel Rodrigues 35Q5335B1035B3535P10modulationperiodic traveling wavesKuramoto-Sivashinsky equationsKorteweg-de Vries equationsBloch decomposition 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 of the Korteweg-de Vries modulation equations supplemented with a source term: relaxation limit in such a system provides, in turn, some stability criteria. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.4955 10.1512/iumj.2013.62.4955 en Indiana Univ. Math. J. 62 (2013) 753 - 783 state-of-the-art mathematics http://iumj.org/access/