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

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.