For my research project I have been investigating the solution on Wikipedia. have read the paper it cites as a reference, by AM Kamchatnov, called Topological Solitons in Magnetohydrodynamics. In that paper Kamchatnov does not refer to Navier-Stokes. He does construct a vector potential for a magnetic field he wants, and that vector potential is taken to be the velocity field in the solution on wikipedia.
However, Kamchatnov does not use that vector field as the velocity field for a fluid himself, nor does he mention, in any form, the pressure or density functions that are listed in the wikipedia solution. Moreover, based on trying to solve the Navier-Stokes equations using the wikipedia functions, it seems as though there is an additional assumption that the energy function is of the form
$$ \rho^3 = C p $$
Where $\rho$ is density, $C$ is a constant, and $p$ is pressure.
With these assumptions, along with body forces and viscosity set to zero, the equations on wikipedia do indeed solve Navier-Stokes. However since Kamchatnov makes no reference to any of this, and only incidentally derives the vector field which wikipedia cherry picks for use as a velocity field, I am currently at a loss as to how $\rho$ and $p$ were actually found in the first place!
I have googled endlessly and asked everyone I know, but everything points back to that Kamchatnov paper, in which I find no mention of what I need to know.
My Question: Does anyone know either where I can find an explanation of where $\rho$ and $p$ come from and why this field was taken as a velocity field in the first place, or can someone explain it?
Thank you.