Ernst equation
In general relativity, the Ernst equation is an integrable non-linear [partial differential equation], named after the American physicist who published it in 1968.
Description
The Ernst's equation governing the complex scalar function is given bywhere is the two-dimensional gradient operator with axisymmetry; for instance, if, then
and if , then
where is the real part of. If is a solution of the Ernst's equation, then is also a solution where is an arbitrary real constant. The transformation belongs to the so-called Ehler's transformation.
Often, one introduces
so that we have
The Ernst equation is derivable from the Lagrangian density
For its Lax pair and other features see e.g. and references therein.