Multilump Symmetric and Nonsymmetric Gravity-Capillary Solitary Waves in Deep Water

Abstract

Multilump gravity-capillary solitary waves propagating in a fluid of infinite depth are computed numerically. The study is based on a weakly nonlinear and dispersive partial differential equation (PDE) with weak variations in the spanwise direction, a model derived by Akers and Milewski [Stud. Appl. Math., 122 (2009), pp. 249--274]. For a two-dimensional fluid, this model agrees qualitatively well with the full Euler equations for the bifurcation curves, wave profiles, and dynamics of solitary waves. Fully localized solitary waves are then computed for three-dimensional fluids. New symmetric lump solutions are computed by using a continuation method to follow the branch of elevation waves. It is then found that the branch of elevation waves has multiple turning points from which new solutions, consisting of multiple lumps separated by smaller oscillations, bifurcate. Nonsymmetric solitary waves, which also feature a multilump structure, are computed and found to appear via spontaneous symmetry-breaking bi...