Fully localised three-dimensional gravity-capillary solitary waves on water of infinite depth

Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence for water of finite depth has recently been established, and in this talk I present an existence theory for water of infinite depth. The governing equations are reduced to a perturbation of the two-dimensional nonlinear Schrödinger equation, which admits a family of localised solutions. Two of these solutions are symmetric in both horizontal directions and an application of a suitable version of the implicit-function theorem shows that they persist under perturbations.
This is joint work with B. Buffoni (EPFL) and E. Wahlén (Lund).