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 this 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.