Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem

A model equation derived by B. B. Kadomtsev & V. I. Petviashvili (1970) suggests 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 spatial direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. The theory is variational in nature. A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties using a generalisation of the Lyapunov-Schmidt reduction procedure. A nontrivial critical point of the reduced functional is detected using the direct methods of the calculus of variations.