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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.

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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