Ultrahyperbolic equation explained

In the mathematical field of differential equations, the ultrahyperbolic equation is a partial differential equation (PDE) for an unknown scalar function of variables of the form

$$\backslash frac\; +\; \backslash cdots\; +\; \backslash frac\; -\; \backslash frac\; -\; \backslash cdots\; -\; \backslash frac\; =\; 0.$$

More generally, if is any quadratic form in variables with signature, 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 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.

References

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

Notes and References

1. See Courant and Hilbert.
2. Web site: On determinism and well-posedness in multiple time dimensions . Proc. R. Soc. A vol. 465 no. 2110 3023-3046 (2008) . Craig . Walter . Weinstein . Steven . 5 December 2013.
3. Elsevier . 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 . 10.1016/j.padiff.2022.100280 . 5 . 100280 . 100280 . 36159725 . 9494226 . free .
4. Springer . 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 . 10.1007/s00521-021-06789-8 . 34 . 8 . 6377–6396. 35936508 . 9355340 .
5. Institut Mittag-Leffler . Helgason, S . 1959. Differential operators on homogeneous spaces . Acta Mathematica . 10.1007/BF02564248 . 102 . 3–4 . 239–299 . free .