Numerical and perturbative computations of solitary waves of the Benjamin-Ono equation with higher order nonlinearity using Christov rational basis functions

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/vertical bar x vertical bar(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 - delta)u(2) + delta u(3) = 0, delta is an element of [0,1] and solved it in two ways. The first was continuation in the homotopy parameter delta, marching from the known Benjamin-Ono soliton for delta = 0 to the cubically-nonlinear soliton delta = 1. The second strategy was to bypass continuation by numerically computing perturbation series in delta and forming Fade approximants to obtain a very accurate approximation at delta = 1. To further minimize computations, we derived an elementary theorem to reduce the two-parameter soliton family to a parameter-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 Korteweg-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. (C) 2011 Elsevier Inc. All rights reserved.