Steady periodic solutions of a variable coefficient KS-KdV equation

A variable coefficient Kuramoto-Sivashinsky-Korteweg-de Vries (KS-KdV) equation is considered that models thin liquid film flow over a periodically corrugated incline. Here, the behaviour of steady, periodic solutions is investigated in the KdV limit with a multi-scale perturbation approach. By applying periodicity constraints, at second order differential equations are derived that describe the slowly varying properties of the first order cnoidal wave solution. The critical points of this system, which correspond to periodic solutions of the governing equation, are determined, and then the spatial stability of this family of solutions is examined. Using Floquet theory, stability criterion are derived, which are then validated with numerical investigations. These numerics also reveal stable limit cycles.