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 13) has recently been given. In this article we present an existence theory for the physically more realistic case 0<β<13. A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey–Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.