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

• Authors: John P. Boyd;Zhengjie Xu
• Venue: Journal of Computational Physics
• Year: 2012

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/|x|^2). To solve the integro-differential equation for waves traveling at a phase speed c, we introduced the artificial homotopy H(u"X"X)-c u+(1-@d)u^2+@du^3=0, @d@?[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 Pade 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 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.