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For a sequence of values of the totalhorizontal impulse that converges to $0$, there are solitarywaves that minimise the energy in a given neighbourhoodof the origin in $W^{2,2}({\mathbb R})$.The problem arises in the frameworkof the classical Euler equation when a two-dimensionallayer of waterabove 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 isassumed to be sub-critical ($ On the energetic stability of solitary water waves , Philos. Trans. Roy. Soc.London Ser. A 360 (2002), 2337–2358]) andby the author ([ Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation , Arch. Rational Mech. Anal.]). Like in the latter, the method is basedon direct minimisation and concentrated compactness, but withoutrelying on "strict sub-additivity", which is still unsettledin the present case. Instead, a complete and careful analysis of minimising sequencesis performed that allows us to reach a conclusion, based only on thenon-existence of "vanishing" minimising sequences.However, in contrast with [ Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation , Arch. Rational Mech. Anal.], we are unable to prove the existenceof minimisers for all small values of the total horizontal impulse.In fact more is needed to get stability, namely that every minimisingsequence has a subsequence that converges to a global minimiser, afterpossible shifts in the horizontal direction. This will be obtainedas a consequence of the analysis of minimising sequences.Then exactly the same argument as in [ Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation , Arch. Rational Mech. Anal.] gives conditional energeticstability and is thereforenot repeated.

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