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Fully localised solitary-wave solutions of the three-dimensional gravity – capillary water-wave problem. (English) Zbl 1133.76010
Summary: The model Kadomtsev-Petviashvili equation 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 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. The reduced functional is related to the functional associated with Kadomtsev-Petviashvili equation, and a nontrivial critical point is detected using the direct methods of the calculus of variations.

MSC:
76B25 Solitary waves for incompressible inviscid fluids
76B45 Capillarity (surface tension) for incompressible inviscid fluids
76M30 Variational methods applied to problems in fluid mechanics
35Q51 Soliton equations
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