On gravity–capillary lumps

Two-dimensional (plane) solitary waves on the surface of water are known to bifurcate from linear sinusoidal wavetrains at specific wavenumbers $k\,{=}\,k_{0}$ where the phase speed $c(k)$ attains an extremum $(\left. \hbox{d}c/\hbox{d}k \right |_{0}\,{=}\,0)$ and equals the group speed. In particular, such an extremum occurs in the long-wave limit $k_{0}\,{=}\,0$, furnishing the familiar solitary waves of the Korteweg–de Vries (KdV) type in shallow water. In addition, when surface tension is included and the Bond number $B\,{=}\,T/(\rho gh^2)\,{<}\,1/3$ ($T$ is the coefficient of surface tension, $\rho$ the fluid density, $g$ the gravitational acceleration and $h$ the water depth), $c(k)$ features a minimum at a finite wavenumber from which gravity–capillary solitary waves, in the form of wavepackets governed by the nonlinear Schrödinger (NLS) equation to leading order, bifurcate in water of finite or infinite depth. Here, it is pointed out that an entirely analogous scenario is valid for the bifurcation of three-dimensional solitary waves, commonly referred to as ‘lumps’, that are locally confined in all directions. Apart from the known lump solutions of the Kadomtsev–Petviashvili I equation for $B\,{>}\,1/3$ in shallow water, gravity–capillary lumps, in the form of locally confined wavepackets, are found for $B\,{<}\,1/3$ in water of finite or infinite depth; like their two-dimensional counterparts, they bifurcate at the minimum phase speed and are governed, to leading order, by an elliptic–elliptic Davey–Stewartson equation system in finite depth and an elliptic two-dimensional NLS equation in deep water. In either case, these lumps feature algebraically decaying tails owing to the induced mean flow.