Conditional energetic stability of gravity solitary waves in the presence of weak surface tension

Abstract

For a sequence of values of the total horizontal impulse that converges to $0$, there are solitary waves that minimise the energy in a given neighbourhood of the origin in $W^{2,2}({\mathbb R})$. The problem arises in the framework of the classical Euler equation when a two-dimensional layer of water above an infinite horizontal bottom is considered, at the surface of which solitary waves propagate under the action of gravity and {\it weak} surface tension. The adjective ``weak'' refers to the Bond number, which is assumed to be sub-critical ($< 1/3$). This extends previous results on the conditional energetic stability of solitary waves in the super-critical case, namely those by A. Mielke ([< i> On the energetic stability of solitary water waves< /i> , Philos. Trans. Roy. Soc. London Ser. A < b> 360< /b> (2002), 2337–2358]) and by the author ([< i> Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation< /i> , Arch. Rational Mech. Anal.]). Like in the latter, the method is based on direct minimisation and concentrated compactness, but without relying on "strict sub-additivity", which is still unsettled in the present case. Instead, a complete and careful analysis of minimising sequences is performed that allows us to reach a conclusion, based only on the non-existence of "vanishing" minimising sequences. However, in contrast with [< i> Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation< /i> , Arch. Rational Mech. Anal.], we are unable to prove the existence of minimisers for < i> all< /i> small values of the total horizontal impulse. In fact more is needed to get stability, namely that every minimising sequence has a subsequence that converges to a global minimiser, after possible shifts in the horizontal direction. This will be obtained as a consequence of the analysis of minimising sequences. Then exactly the same argument as in [< i> Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation< /i> , Arch. Rational Mech. Anal.] gives conditional energetic stability and is therefore not repeated.