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 (dc/dk| 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/(ρgh 2 ) 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.