Resonant sloshing in shallow water

Abstract

The ordinary differential equation \[ {\textstyle\frac{1}{3}}\kappa^2(g^{\prime\prime}+g) - \lambda g - {\textstyle\frac{3}{2}}g^2 + \frac{2}{\pi} \cos t = -\frac{3}{2}\int_{-\pi}^{\pi}g^2\,{\rm d}t, \] which represents forced water waves on shallow water near resonance, is considered when the dispersion κ is small. Asymptotic methods are used to show that there are multiple solutions with period 2π for a given value of the detuning parameter λ. The effects of dissipation are also considered.