Existence and energetic stability of 3D fully-localized gravity capillary waves

equation The paper “Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves“, co-authored by B. Buffoni, M. Groves, S. Sun and E. Wahlen, has been published in the Journal of Differential Equations. They show that the hydrodynamic problem for 3D water waves with strong surface tension admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction.  The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy subject to the constraint that the momentum is fixed.  Minimisation then implies stability.  “Stability” is qualified due to the lack of global well-posedness for 3D water waves. A link to the paper is here.

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