Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves

We prove the existence of a minimiser of E subject to the constraint I=2μ, where 0<μ≪1. The existence of a small-amplitude solitary wave is thus assured, and since E and I are both conserved quantities a standard argument may be used to establish the stability of the set Dμ of minimisers as a whole. 'Stability' is however understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves. We show that solutions to the evolutionary problem starting near Dμ remain close to Dμ in a suitably defined energy space over their interval of existence; they may however explode in finite time due to higher-order derivatives becoming unbounded.