Ernst equation

Not to be confused with Nernst equation.

In mathematics, the Ernst equation^[1] is an integrable non-linear partial differential equation, named after the American physicist Frederick J. Ernst [Wikidata].^[2]

The Ernst equation

The equation reads: $${\displaystyle \Re (u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}.}$$

where ${\textstyle \Re (u)}$ is the real part of ${\textstyle u}$. For its Lax pair and other features see e.g. ^[3][4] and references therein.

Usage

The Ernst equation is employed in order to produce exact solutions of the Einstein's equations in the general theory of relativity.