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.

A theoretical study is made of fully localized solitary waves, commonly referred to as ‘lumps’, on the interface of a two-layer fluid system in the case that the upper layer is bounded by a rigid lid and lies on top of an infinitely deep fluid. The analysis is based on an extension, that allows for weak transverse variations, of the equation derived by Benjamin (J. Fluid Mech. vol. 245, 1992, p. 401) for the evolution in one spatial dimension of weakly nonlinear long waves in this flow configuration, assuming that interfacial tension is large and the two fluid densities are nearly equal. The phase speed of the Benjamin equation features a minimum at a finite wavenumber where plane solitary waves are known to bifurcate from infinitesimal sinusoidal wavetrains. Using small-amplitude expansions, it is shown that this minimum is also the bifurcation point of lumps akin to the free-surface gravity–capillary lumps recently found on water of finite depth. Numerical continuation of the two symmetric lump-solution branches that bifurcate there reveals that the elevation-wave branch is directly connected to the familiar lump solutions of the Kadomtsev–Petviashvili equation, while the depression-wave branch apparently features a sequence of limit points associated with multi-modal lumps. Plane solitary waves of elevation, although stable in one dimension, are unstable to transverse perturbations, and there is evidence from unsteady numerical simulations that this instability results in the formation of elevation lumps.

Gravity-capillary solitary waves of depression, that bifurcate at the minimum phase speed on water of finite or infinite depth, while stable to perturbations along the propagation direction, are found to be unstable to transverse perturbations on the basis of a long-wave stability analysis. This suggests a possible generation mechanism of the new class of gravitycapillary lumps recently shown to also bifurcate at the minimum phase speed.

Interfacial gravity-capillary plane solitary waves, driven by the gravitational force in the presence of interfacial tension in a two-layer deep-water potential flow, bifurcate in the form of wavepackets with a non-zero carrier wavenumber at which the phase speed is minimized. A stability property for the interfacial gravity-capillary plane solitary waves is presented within the framework of the full Euler equations: according to a linear stability analysis based on the perturbation method, such waves are unstable under weak and long-wave disturbances in the transverse direction to the dominant wave propagation. An instability criterion is verified that the total mechanical energy of the solitary waves is a decreasing function of the solitary wavespeed, owing to the fact that the speed of the bifurcating solitary wavepackets is less than the minimum of the phase speed. This result is consistent with an earlier study on the transverse instability of the longitudinally stable interfacial gravity-capillary solitary waves from the Benjamin model equation for weakly nonlinear long interfacial elevations (Kim and Akylas, J Fluid Mech 557:237-256, 2006). The analysis is also applicable to other interfacial gravity-capillary solitary waves that may bifurcate below the minimum of the phase speed, regardless of any restrictions on fluid depths in two-layer potential flows.

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