A numerical method for the Benjamin-Ono equation

This paper is concerned with the numerical solution of the Cauchy problem for the Benjamin-Ono equationu _t +uu _x −Hu _xx =0, whereH denotes the Hilbert transform. Our numerical method first approximates this Cauchy problem by an initial-value problem for a corresponding 2L-periodic problem in the spatial variable, withL large. This periodic problem is then solved using the Crank-Nicolson approximation in time and finite difference approximations in space, treating the nonlinear term in a standard conservative fashion, and the Hilbert transform by a quadrature formula which may be computed efficiently using the Fast Fourier Transform.