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I know that "soliton" waves can consist of a crest without a trough. One would expect the reverse to be true as well.

However, this Wikipedia excerpt says,

So for this nonlinear gravity wave it is the total water depth under the wave crest that determines the speed, with higher waves traveling faster than lower waves. Note that solitary wave solutions only exist for positive values of H, solitary gravity waves of depression do not exist.

Is this true, and if so, why?

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  • $\begingroup$ Can you give a citation to the specific language in that page? Did they say that equations that produce solitons can not have an almost everywhere nearly constant non-zero solution with a stable "dip" in them? $\endgroup$
    – FlatterMann
    Commented Jun 17, 2023 at 5:40
  • $\begingroup$ A link to the Wikipedia article Dispersion (water waves) in section shallow water. $\endgroup$
    – Farcher
    Commented Jun 17, 2023 at 7:18
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    $\begingroup$ This is a good question and its answer is nontrivial. $\endgroup$
    – hyportnex
    Commented Jun 17, 2023 at 14:16
  • $\begingroup$ From the very deep past what I can recall is that the issue is related to the fact the higher the wave the faster it is, thus a depression cannot be stable. This you can see by neglecting the dispersive term represented by the 3rd derivative $u_{xxx}$ and by writing the remainder of the KdV as $u_t+uu_x$ whose solution is $u=f(s)$ with $s=x-ut$ for any differentiable $f(s)$. This shows that the apparent nondispersive wave speed is $u$. In a trough the surrounding is always at a higher speed than the trough itself, so it cannot be stable. $\endgroup$
    – hyportnex
    Commented Jun 17, 2023 at 19:49

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