A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

Abstract

Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as 'lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number β greater than 1 3 ) has recently been given. In this article we present an existence theory for the physically more realistic case 0 < β <

Integral and Asymptotic Properties of Solitary Waves in Deep Water

“…While there are no rigorous existence results in the literature for three-dimensional gravity-capillary waves in infinite depth, we have been informed by Buffoni, Groves, and Wahlén that their finite-depth construction in [11] can be extended to infinite depth and moreover that there is an alternate construction in this case using the implicit function theorem. Three-dimensional capillary-gravity waves have also been calculated formally [27] and numerically [1,34,41].…”

“…Solitary waves in finite depth, where the fluid is instead bounded below by a flat bed, have a long and celebrated history; see, for instance, the reviews [16,19,33]. This includes a wide variety of existence results for gravity waves in two dimensions [4,5,17,29,32] and gravity-capillary waves in two [3,8,10,24,28] and three [9,11,20] dimensions. The two-dimensional gravity waves are waves of elevation in that their free surface elevations are everywhere positive [14], while some of the gravity-capillary waves are waves of depression with negative free surfaces, and still others have oscillatory free surfaces that change sign.…”

A variational principle for three-dimensional water waves over Beltrami flows

“…In the irrotational case, when ∇ × u = 0 everywhere in Ω, there are several existence results for different types of three-dimensional waves, including doubly-periodic waves, fully localized solitary waves and waves with a solitary-wave profile in one horizontal direction and periodic or quasi-periodic profile in another (see e.g. [7,8,13,20,21,22,25,26,31] and references therein).…”

Small-amplitude fully localised solitary waves for the full-dispersion Kadomtsev–Petviashvili equation

“…The formula I ε = E − cM with c = 1 − ε 2 defines a smooth functional I ε : X → R which has a nontrivial critical point for each sufficiently small value of ε > 0. To motivate our main result it is instructive to review the formal derivation of the (normalised) steady KP equation (5) from the steady FDKP equation (3). We begin with the linear dispersion relation for a two-dimensional sinusoidal travelling wave train with wave number k 1 and speed c, namely…”

“…The function k 1 → c(k 1 ), k 1 ≥ 0 has a unique global minimum at k 1 = 0 with c(0) = 1 (see Figure 2). Bifurcations of nonlinear solitary waves are expected whenever the linear group and phase speeds are equal, so that c (k 1 ) = 0 (see Dias & Kharif [8,§3]); one therefore expects bifurcation of small-amplitude solitary waves from uniform flow with unit speed. Furthermore, observing that m is an analytic function of k 1 and k2 k1 (note that |k|…”