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

*(English)*Zbl 1133.76010Summary: 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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\textit{M. D. Groves} and \textit{S. M. Sun}, Arch. Ration. Mech. Anal. 188, No. 1, 1--91 (2008; Zbl 1133.76010)

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