16 ISSN 0001-4338, Izvestiya, Atmospheric and Oceanic Physics, 2020, Vol. 56, No. 1, pp. 16–32. © Pleiades Publishing, Ltd., 2020. Russian Text © The Author(s), 2020, published in Izvestiya Rossiiskoi Akademii Nauk, Fizika Atmosfery i Okeana, 2020, Vol. 56, No. 1, pp. 20–42. Nonlinear Waves in a Rotating Ocean (The Ostrovsky Equation and Its Generalizations and Applications) Y. A. Stepanyants a, b, * a School of Sciences, University of Southern Queensland, Toowoomba, QLD, 4350 Australia b Department of Applied Mathematics, Alekseev Technical University, Nizhny Novgorod, 603950 Russia *e-mail: Yury.Stepanyants@usq.edu.au Received September 11, 2019; revised September 25, 2019; accepted September 25, 2019 Abstract—This review presents theoretical, numerical, and experimental results of a study of the structure and dynamics of weakly nonlinear internal waves in a rotating ocean accumulated over the past 40 years since the time when the approximate equation, called the Ostrovsky equation, was derived in 1978. The relationship of this equation with other well-known wave equations, the integrability of the Ostrovsky equation, and the con- dition for the existence of stationary solitary waves and envelope solitary waves are discussed. The adiabatic dynamics of Korteweg–de Vries solitons in the presence of fluid rotation is described. The mutual influence of the ocean inhomogeneity and rotation effect on the dynamics of solitary waves is considered. The univer- sality of the Ostrovsky equation as applied to waves in other media (solids, plasma, quark–gluon plasma, and optics) is noted. Keywords: surface and internal waves, rotating fluid, solitary waves, solitons, integrable equations DOI: 10.1134/S0001433820010077 INTRODUCTION The role of the Korteweg–de Vries (KdV) equation in the description of nonlinear wave phenomena in weakly dispersive media is well known. This equation successfully combines the competition of the main effects in the physics of nonlinear waves, dispersion and nonlinearity. After numerous works in the 1950s- 1960s, mainly in plasma physics, which led to the der- ivation of the KdV equation in these media, the uni- versal role of this equation was realized and, through the numerical simulation, the formation of solitary waves (called solitons) was discovered from arbitrary initial perturbations [1, 2]. Further interest in this equation led to the discovery of a method for solving a wide class of nonlinear equations - the inverse scatter- ing transform and its generalizations [3–5]. Shortly afterwards, related equations were found in the nonlinear wave theory, such as the modified KdV equation (mKdV) describing waves in media with cubic nonlinearity, the Gardner equation containing combined quadratic and cubic nonlinearities (espe- cially popular in the theory of internal waves in the ocean), the Benjamin–Ono (BO) and Joseph– Kubota–Ko–Dobbs equations describing internal waves in a deep ocean [5], the Kadomtsev–Petviash- vili equation (KP) [6, 7] that described multidimen- sional effects associated with the transverse diffraction of wave beams, etc. The Ostrovsky equation, published in 1978 [8] and describing weakly nonlinear wave pro- cesses in the ocean taking into account the Earth rota- tion, can be put on the same list. This equation general- izes the KdV equation by the inclusion of an additional dispersion term and has the following form: (1) Here η denotes the shift of a free surface in the case of surface waves or a pycnocline perturbation in the case of internal waves. The equation coefficients are expressed as follows via the hydrological parameters of the problem: for surface waves: (2) where h is the depth of the basin and g is the accelera- tion of gravity; f = 2Ω sinϕ is the Coriolis parameter that includes the Earth rotation frequency Ω and the geographic latitude of the location ϕ. for internal waves in a two-layer fluid in the Boissinesq approximation: (3)   ∂η ∂η ∂η ∂η ∂ + + αη +β = γη   ∂ ∂ ∂ ∂ ∂   3 3 . c x t x x x = α= β= γ= 2 2 , 2 6 2 3 , , , h c f c ch c gh - δρ = α= ρ + β= γ= 1 2 1 2 1 2 1 2 2 1 2 3 , , 2 , , 6 2 hh h h c c g h h hh chh f c