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I'm now reading papers about the the well-posedness of Euler and Navier–Stokes equation, so I wonder if we have soliton solutions for these two equations just like for KdV equation. I'm interested in this because if soliton solutions exist, then we can try larger space for initial data, which includes the soliton, to work in for the well-posedness, and also we can consider the stability for the soliton solutions.

I searched in google, but haven't got any positive result.

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There are solitary wave solutions for the Euler equations, but they do not have the "soliton" property of passing through each other without changing shape. Friedrichs and Hyers proved existence of such solutions in the 1950s for the case of zero surface tension. The problem with surface tension was solved in the 1980s and 1990s.

Here is one reference, which will lead you to the earlier ones:

S.M. Sun, Proc. Roy. Soc. London A 455 (1999), 2191-2228.

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    $\begingroup$ A bit more info on the Sun paper, "Nonexistence of truly solitary waves in water with small surface tension." From the Abstract: "It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals." jstor.org/pss/53320 $\endgroup$
    – Joseph O'Rourke
    Commented Aug 16, 2011 at 10:37
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Yes! There exist soliton solutions for the Navier Stokes d.e. (and thus also for the Euler d.e), these has been recently found. Please look at the following citation; r. meulens, "A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation," AIP Advances 12, vol. 12 (015308), no. https://doi.org/10.1063/5.0074083, p. 11, 2022.

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    $\begingroup$ Does this disagree with @MichaelRenardy's answer? $\endgroup$
    – LSpice
    Commented Nov 10, 2022 at 21:21

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