### Gravity solitary waves by minimization: an uncountable family

#### Abstract

We improve and simplify the minimization method for solitary

waves in two cases: firstly, when the surface tension is weak (that is,

the Bond number is $< 1/3$) and the depth is finite, and secondly,

when the depth is infinite.

In a previous work on the first case,

minimizers were shown to exist for a sequence tending to $0$

of values of the horizontal impulse. The main difficulty

is that strict subadditivity in the concentration-compactness

method is unsettled.

Here we observe in both examples

that strict subadditivity nevertheless holds for

a set of horizontal impulses of positive measure and the related

propagation speeds are estimated from above.

waves in two cases: firstly, when the surface tension is weak (that is,

the Bond number is $< 1/3$) and the depth is finite, and secondly,

when the depth is infinite.

In a previous work on the first case,

minimizers were shown to exist for a sequence tending to $0$

of values of the horizontal impulse. The main difficulty

is that strict subadditivity in the concentration-compactness

method is unsettled.

Here we observe in both examples

that strict subadditivity nevertheless holds for

a set of horizontal impulses of positive measure and the related

propagation speeds are estimated from above.

#### Keywords

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

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