Ernst equation
# Ernst equation

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

## The Ernst equation

The equation reads:

$\Re (u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}.$

For its Lax pair and other features see e.g. and references therein.

### Usage

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

