Weakly non-local solitons for capillary-gravity waves: Fifth-degree Korteweg-de Vries equation

Abstract

Hunter and Scheurle have shown that capillary-gravity water waves in the vicinity of Bond number (Bo)[approximate]1/3 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 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 [epsilon]. 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.