Ernst equation

In mathematics, the Ernst equation is an integrable non-linear partial differential equation, named after the American physicist Frederick J. Ernst.

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. 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.

References