# Ernst equation

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

## The Ernst equation

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

For its Lax pair and other features see e.g. [3], [4] and references therein.

### Its usage

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

## References

1. ^ Weisstein, Eric W, Ernst equation, MathWorld--A Wolfram Web.
2. ^ Biography of Frederick J. Ernst
3. ^ Harrison B.K. (1978), Backlund transformation for the Ernst equation of general relativity, Phys. Rev. Lett. 41, no. 18, 1197--1200.
4. ^ Marvan M. (2004), Recursion operators for vacuum Einstein equations with symmetries, in: Symmetry in nonlinear mathematical physics, Kyiv.arXiv:nlin/0401014