Five Regimes of the Quasi-Cnoidal, Steadily Translating Waves of the Rotation-Modified Korteweg-deVries (“Ostrovsky”) Equation

The Rotation-Modified Korteweg-deVries (RMKdV) equation differs from the ordinary KdV equation only through an extra undifferentiated term due to Coriolis force. This article describes the steadily-travelling, spatially-periodic solutions which have peaks of identical size. These generalize the “cnoidal” waves of the KdV equation. There are five overlapping regimes in the parameter space. We derive four different analytical approximations to interpret them. There is also a narrow region where the solution folds over so that there are three distinct shapes at a given point in parameter space. The shortest of these shapes is approximated everywhere in space by a parabola except for a thin interior layer at the crest. A low-order Fourier-Galerkin algorithm, usually thought of as a numerical method, also yields an explicit analytic approximation, too. We illustrate the usefulness of these approximations through comparisons with pseudospectral Fourier numerical computations over the whole parameter space.