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

• A multi-scale perturbation solution of the steady fKdVB equation is outlined.
• A new solution is derived for the linearised stability of cnoidal waves.
• The fixed point solutions of the perturbation system are analysed.
• These solutions are compared to experimental results for sloshing in a tank.

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).