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posted on 22.07.2005by Mark D. Groves, S.M. Sun
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.