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.

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. Wang, Y; Shen, Y; Deng, D; Dinov, ID (2022). "Determinism, Well-posedness, and Applications of the Ultrahyperbolic Wave Equation in Spacekime". Partial Differential Equations in Applied Mathematics. 5 (100280). Elsevier: 100280. doi:10.1016/j.padiff.2022.100280. PMC 9494226. PMID 36159725.
4. Zhang, R; Zhang, Y; Liu, Y; Guo, Y; Shen, Y; Deng, D; Qiu, Y; Dinov, ID (2022). "Kimesurface Representation and Tensor Linear Modeling of Longitudinal Data". Partial Differential Equations in Applied Mathematics. 34 (8). Springer: 6377–6396. doi:10.1007/s00521-021-06789-8. PMC 9355340. PMID 35936508.
5. Helgason, S (1959). "Differential operators on homogeneous spaces". Acta Mathematica. 102 (3–4). Institut Mittag-Leffler: 239–299. doi:10.1007/BF02564248.

References

• Richard Courant; David Hilbert (1962). Methods of Mathematical Physics, Vol. 2. Wiley-Interscience. pp. 744–752. ISBN 978-0-471-50439-9.
• Lars Hörmander (20 August 2001). "Asgeirsson's Mean Value Theorem and Related Identities". Journal of Functional Analysis. 2 (184): 377–401. doi:10.1006/jfan.2001.3743.
• Lars Hörmander (1990). The Analysis of Linear Partial Differential Operators I. Springer-Verlag. Theorem 7.3.4. ISBN 978-3-540-52343-7.
• Sigurdur Helgason (2000). Groups and Geometric Analysis. American Mathematical Society. pp. 319–323. ISBN 978-0-8218-2673-7.
• Fritz John (1938). "The Ultrahyperbolic Differential Equation with Four Independent Variables". Duke Math. J. 4 (2): 300–322. doi:10.1215/S0012-7094-38-00423-5.