### 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.

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.

#### Keywords

Capilary-gravity water waves; solitary waves; stability; variational methods

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