Modulation theory for the steady forced KdV–Burgers equation and the construction of periodic solutions

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg–de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger’s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).