an:05261598
Zbl 1133.76010
Groves, M. D.; Sun, S.-M.
Fully localised solitary-wave solutions of the three-dimensional gravity -- capillary water-wave problem
EN
Arch. Ration. Mech. Anal. 188, No. 1, 1-91 (2008).
0003-9527 1432-0673
2008
j
76B25 76B45 76M30 35Q51
variational principle; Kadomtsev-Petviashvili equation; critical point
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.