Wave Motion 35 (2002) 141–155 Five regimes of the quasi-cnoidal, steadily translating waves of the rotation-modified Korteweg-de Vries (“Ostrovsky”) equation John P. Boyd a,∗ , Guan-Yu Chen b a Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109, USA b Institute of Harbour and Marine Technology, Wuchi, Taichung 435, Taiwan Received 23 August 1999; received in revised form 27 March 2001; accepted 2 April 2001 Abstract The rotation-modified Korteweg-de Vries (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. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Solitary wave; Cnoidal wave; Water waves 1. Introduction The rotation-modified Korteweg-de Vries (RMKdV) equation, which is often called the “Ostrovsky” equation, is a model for gravity waves propagating down a channel under the influence of Coriolis force: ∂ ∂x (u t + uu x + u xxx ) - ǫ 2 u = 0 RMKdV equation, (1) where the nondimensional parameter ǫ ≥ 0 measures the importance of the earth’s rotation. In the limit ǫ = 0, this equation reduces to the ordinary Korteweg-de Vries equation. The derivation and 20 year history of this equation are discussed at length in [4,5,8,12,13]. The steadily translating solutions of the RMKdV equation solve the ordinary differential equation -cu XX + u 2 X + uu XX + u 4X - ǫ 2 u = 0 stationary RMKdV, (2) where X = x - ct and c is the phase speed. (Note that by integrating this term-by-term, one proves that if u is periodic with period P (including the limit P →∞), then  P 0 u dX = 0). The ODE form of the RMKdV equation, ∗ Corresponding author. E-mail addresses: jpboyd@engin.umich.edu (J.P. Boyd), gychen@mail.ihmt.gov.tw (G.-Y. Chen). 0165-2125/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S0165-2125(01)00097-X