Ernst equation explained

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

The Ernst equation

The equation reads:$$\backslash Re(u)(u\_+u\_r/r+u\_)\; =\; (u\_r)^2+(u\_z)^2.$$

where $\backslash Re(u)$ is the real part of $u$. 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.

References

1. http://mathworld.wolfram.com/ErnstEquation.html Weisstein, Eric W, Ernst equation, MathWorld--A Wolfram Web.
2. Web site: Biography of Frederick J. Ernst . 2017-05-09 . https://web.archive.org/web/20180104233406/http://mypages.iit.edu/~segre/iit_physics_bios/ernst_f.html . 2018-01-04 . dead .
3. Harrison . B. Kent . Bäcklund Transformation for the Ernst Equation of General Relativity . Physical Review Letters . American Physical Society (APS) . 41 . 18 . 30 October 1978 . 0031-9007 . 10.1103/physrevlett.41.1197 . 1197–1200. 1978PhRvL..41.1197H .
4. Marvan . M. . 2004 . Recursion operators for vacuum Einstein equations with symmetries . Proceedings of the Conference on Symmetry in nonlinear mathematical physics. Kyiv, Ukraine. nlin/0401014. Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine. 50. 179–183.