In the mathematical field of differential equations, the ultrahyperbolic equation is a partial differential equation (PDE) for an unknown scalar function u of 2n variables x1, ..., xn, y1, ..., yn of the form
More generally, if a is any quadratic form in 2n variables with signature (n, n), then any PDE whose principal part is
Any such equation can be put in the form 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.
In 2008, Walter Craig and Steven Weinstein proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one hypersurface.
[2] And later, in 2022, a research team at the University of Michigan extended the conditions for solving ultrahyperbolic wave equations to complex-time (kime), demonstrated space-kime dynamics, and showed data science applications using tensor-based linear modeling of functional magnetic resonance imaging data.
[3][4] The equation has also been studied from the point of view of symmetric spaces, and elliptic differential operators.
[5] In particular, the ultrahyperbolic equation satisfies an analog of the mean value theorem for harmonic functions.