Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves

In this paper we show that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction. The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy, which is given in dimensionless coordinates by epsilon(eta, phi) = integral(R2){1/2 integral(1+eta)(0) (phi(2)(x) + phi(2)(y) + phi(2)(z))dy +1/2 eta(2) + beta[root 1 + eta(2)(x) + eta(2)(z) - 1]}dxdz, subject to the constraint that the momentum I(eta, phi) = integral(R2)eta(x)phi vertical bar(y=1+eta)dzdz is fixed; here {(x, y, z): x, z is an element of R, y is an element of (0, 1 + eta(x, z))} is the fluid domain, phi is the velocity potential and beta > 1/3 is the Bond number. These functionals are studied locally for eta in a neighbourhood of the origin in H-3(R-2). We prove the existence of a minimiser of epsilon subject to the constraint I = 2 mu, where 0 < mu << 1. The existence of a small-amplitude solitary wave is thus assured, and since epsilon and I are both conserved quantities a standard argument may be used to establish the stability of the set D-mu of minimisers as a whole. 'Stability is however understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves. We show that solutions to the evolutionary problem starting near D-mu remain close to D-mu in a suitably defined energy space over their interval of existence; they may however explode in finite time due to higher-order derivatives becoming unbounded. (C) 2012 Elsevier Inc. All rights reserved.