Weakly non-local solutions for capillary-gravity waves: fifth-degree Korteweg-de Vries equation

Hunter and Scheurle have shown that capillary-gravity water waves in the vicinity of Bond number (Bo) = l/3 are consistently modelled by the Korteweg-de Vries equation with the addition of a fifth derivative term. This wave equation does not have strict soliton solutions for Bo < l/3 because the near-solitons have oscillatory “wings” that extend indefinitely from the core of the wave. However, these solutions are “arbitrarily small perturbations of solitary waves” because the amplitude of the “wings” is exponentially small in the amplitude 6 of the “core”. Pomeau, Ramani, and Grammaticos have calculated the amplitude of the “wings” by applying matched asymptotics in the complex plane in the limit E + 0. In this article, we describe a mixed Chebyshev/radiation function pseudospectral method which is able to calculate the “weakly non-local solitons” for all E. We show that for fixed phase speed, the solitons form a three-parameter family because the linearized wave equation has three eigensolutions. We also show that one may repeat the soliton with even spacing to create a three-parameter of periodic solutions, which we also compute. Because the amplitude of the “wings” is exponentially small, these non-local capillary gravity solitons are as interesting as the classical, localized solitons that solve the Korteweg-de Vries equation.