# 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

${\displaystyle {\frac {\partial ^{2}u}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}u}{\partial x_{n}^{2}}}-{\frac {\partial ^{2}u}{\partial y_{1}^{2}}}-\cdots -{\frac {\partial ^{2}u}{\partial y_{n}^{2}}}=0.\qquad \qquad (1)}$

More generally, if a is any quadratic form in 2n variables with signature (n,n), then any PDE whose principal part is ${\displaystyle a_{ij}u_{x_{i}x_{j}}}$ is said to be ultrahyperbolic. Any such equation can be put in the form 1. above by means of a change of variables. [1]

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. [2]

The equation has also been studied from the point of view of symmetric spaces, and elliptic differential operators. [3] In particular, the ultrahyperbolic equation satisfies an analog of the mean value theorem for harmonic functions.

## Notes

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.

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