Stability of Deep Water Waves Governed by the Benjamin--Ono Equation

E. Infeld Sołtan Institute for Nuclear Studies, Hoża 69, 00-681 Warsaw, Poland and G. Rowlands Department of Physics, University of Warwick, Coventry CV4 7AL, UK

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Received: September 30, 2002

The Benjamin--Ono equation models the dynamics of internal waves in stratified fluids of great depth. It includes an integral (Hilbert transform) term, and so stability calculations might seem difficult. We expand in both the amplitude of the nonlinear wave and the wave vector of the perturbation, assumed to be small quantities of the same order. An expression for the nonlinear dispersion relation is obtained. Nonlinear periodic Benjamin--Ono waves are stable, just as the localized, algebraic soliton solutions (Lorentzians), already known to be stable. (This also follows as a limit of our calculations.) We extend the known analogy between the Benjamin--Ono and modified Korteweg--de Vries equations.

DOI: 10.12693/APhysPolA.103.365 PACS numbers: 47.20.Ky, 52.35.Py