# Ultrahyperbolic equation

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In the mathematical field of partial differential equations, the ultrahyperbolic equation is a partial differential equation for an unknown scalar function u of 2n variables x1, ..., xn, y1, ..., yn of the form

${\frac {\partial ^{2}u}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}u}{\partial x_{n}^{2}}}-{\frac {\partial ^{2}u}{\partial y_{1}^{2}}}-\cdots -{\frac {\partial ^{2}u}{\partial y_{n}^{2}}}=0.\qquad \qquad (1)$ More generally, if a is any quadratic form in 2n variables with signature (n,n), then any PDE whose principal part is $a_{ij}u_{x_{i}x_{j}}$ is said to be ultrahyperbolic. Any such equation can be put in the form 1. above by means of a change of variables. 

The ultrahyperbolic equation has been studied from a number of viewpoints. On the one hand, it resembles the classical wave equation. This has led to a number of developments concerning its characteristics, one of which is due to Fritz John: the John equation.

Walter Craig and Steven Weinstein recently (2008) proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one hypersurface. 

The equation has also been studied from the point of view of symmetric spaces, and elliptic differential operators.  In particular, the ultrahyperbolic equation satisfies an analog of the mean value theorem for harmonic functions.

1. See Courant and Hilbert.
2. Craig, Walter; Weinstein, Steven. "On determinism and well-posedness in multiple time dimensions". Proc. R. Soc. A vol. 465 no. 2110 3023-3046 (2008). Retrieved 5 December 2013.
3. See, for instance, Helgasson.