Modulation theory for the Korteweg-de Vries equation with damping and periodic forcing

Abstract

The Korteweg-de Vries (KdV) equation governs the evolution of weakly nonlinear and weakly dispersive long waves in a wide variety of applications. We first present asymptotic solutions to the KdV equation, perturbed by Burgers damping and periodic forcing. This equation models, amongst other applications, the resonant forcing of shallow water waves in a container. In particular, we seek periodic solutions to the steady forced KdV-Burgers (fKdVB) equation using a multiple-scale perturbation approach, where the first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Then using the second order equation in the hierarchy, we find a system of differential equations describing the slowly varying properties of the cnoidal wave. The fixed point solutions of this system are analysed, which correspond to periodic solutions to the fKdVB equation that are fully defined at first and second order. Furthermore, the stability of these solutions is established by conducting a linear stability analysis with this system of differential equations about the fixed points. As well, to support these findings, Floquet theory is used to determine stability. The unsteady fKdVB equation is also considered, where a multiple-scale perturbation technique based on modulated cnoidal waves is again applied. From the second order equation, we arrive at the well known `Whitham equations' with additional terms attributed to the damping and forcing. Next, steady solutions are sought using these modulation equations and as a result, the same family of first order steady periodic solutions that were previously identified, are now found. Moreover, our analysis is extended to the steady forced Kuramoto-Sivashinsky equation, which describes thin film flow down an inclined, corrugated surface. Subsequently, steady periodic solutions at first and second order are derived for this new problem and their stability is investigated.