On solitary waves. Part 2. A unified perturbation theory for higher-order waves

Abstract

A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on (i) the accuracy of the theoretically predicted wave properties and (ii) the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property (e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy), on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ(n) (the relative mean-square difference between successive orders n of wave elevations) reaching a minimum, δ_m, at a specific order, n=n_m, both depending on the base adopted, e.g. n_(m, α)=11-12 based on parameter α (wave amplitude), n_(m, β)=15 on β (amplitude-speed square ratio), and n_(m, ∈)=17 on ∈ (wave number squared). The asymptotic range is brought to completion by the highest order of n=18 reached in this work.