Numerical and perturbative computations of solitary waves of the Benjamin–Ono equation with higher order nonlinearity using Christov rational basis functions John P. Boyd a,⇑ , Zhengjie Xu b a Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109-2143, United States b Program in Applied and Interdisciplinary Mathematics, University of Michigan, East Hall, Ann Arbor, MI 48109, United States article info Article history: Received 14 May 2011 Received in revised form 13 September 2011 Accepted 2 October 2011 Available online 20 October 2011 Keywords: Benjamin–Ono equation Pseudospectral method Christov orthogonal rational functions Rational basis functions Pseudospectral Soliton Solitary wave abstract Computation of solitons of the cubically-nonlinear Benjamin–Ono equation is challenging. First, the equation contains the Hilbert transform, a nonlocal integral operator. Second, its solitary waves decay only as O(1/jxj 2 ). To solve the integro-differential equation for waves traveling at a phase speed c, we introduced the artificial homotopy H(u XX )  c u + (1  d)u 2 + du 3 = 0, d 2 [0, 1] and solved it in two ways. The first was continuation in the homotopy parameter d, marching from the known Benjamin–Ono soliton for d = 0 to the cubically-nonlinear soliton at d = 1. The second strategy was to bypass continuation by numerically computing perturbation series in d and forming Padé approximants to obtain a very accurate approximation at d = 1. To further minimize computations, we derived an elementary theorem to reduce the two-parameter soliton family to a parame- ter-free function, the soliton symmetric about the origin with unit phase speed. Solitons for higher order Benjamin–Ono equations are also computed and compared to their Kor- teweg–deVries counterparts. All computations applied the pseudospectral method with a basis of rational orthogonal functions invented by Christov, which are eigenfunctions of the Hilbert transform. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction The Benjamin–Ono equation, sometimes called the Benjamin–Davis–Ono equation, was independently derived by Benja- min [6] and Davis and Acrivos [20] through singular perturbation theory as an approximation to weakly nonlinear water waves in deep water: u t þ uu x þ Hðu xx Þ¼ 0 ½Benjamin—Ono equation; ð1Þ where a subscripted coordinate denotes differentiation with respect to that coordinate and H is the Hilbert transform defined by H½f ðxÞ 1 p P Z 1 1 f ðyÞ y  x dy; ð2Þ where P denotes the Cauchy principal value. The solitary waves are [35] 0021-9991/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2011.10.004 ⇑ Corresponding author. E-mail addresses: jpboyd@umich.edu (J.P. Boyd), zhengjie.xu@gmail.com (Z. Xu). Journal of Computational Physics 231 (2012) 1216–1229 Contents lists available at SciVerse ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp