Ultrahyperbolic equation

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, ..., x_n, y₁, ..., y_n of the form

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

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 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.