Integral convergence of the higher-order theory for solitary waves

Abstract

An exact analytic solution for a solitary wave of arbitrary height is attained by series expansions of flow variables based on parameter ε = k²h², (k being the wave number of the solitary wave on water of uniform depth h) by orders in O(ε^n) up to n = 25. Its convergence behavior is found first to yield a set of asymptotic representations for all the flow variables, each and every becoming highest in accuracy at O(ε^(17). For n > 17, the field variables and wave parameters, e.g., wave amplitude, have their errors continue increasing with n, but, in sharp contrast, all the wave integral properties including the excess mass first undergo finite fluctuations from O(ε^(17) to O(ε^(20), then all converge uniformly beyond in a group of tight bundle within the range 0 < ε < 0.283, with ε = 0.283 corresponding to the highest solitary wave with a 120° vertex angle. This remarkable behavior of series convergence seems to have no precedent, and furthermore, is unique in ε, not shared by the exact solutions based on all other parameters examined here.

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