Ultrahyperbolic wave equation

In the mathematical field of partial differential equations, the ultrahyperbolic wave equation is a partial differential equation for an unknown scalar function "u" of 2"n" variables "x"₁, ..., "x"_"n", "y"₁, ..., "y"_"n" of the form

: $frac\{partial^2\; u\}\{partial\; x\_1^2\}+cdots+frac\{partial^2\; u\}\{partial\; x\_n^2\}-frac\{partial^2\; u\}\{partial\; y\_1^2\}-cdots-frac\{partial^2\; u\}\{partial\; y\_n^2\}=0.qquadqquad(1)$

More generally, if "a" is any quadratic form in 2"n" variables with signature ("n","n"), then any PDE whose principal part is $a\_\{ij\}u\_\{x\_ix\_j\}$ is said to be ultrahyperbolic. Any such equation can be put in the form 1. above by means of a change of variables. [See Courant and Hilbert.]

The ultrahyperbolic wave 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.

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

Any constant-coefficient PDE whatsoever can be realized as a reduction of the ultrahyperbolic wave equation in possibly more variables. [See Courant and Hilbert.]

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