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, ..., 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 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. Elsevier. 5 (100280): 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. Springer. 34 (8): 6377–6396. doi:10.1007/s00521-021-06789-8. PMC 9355340. PMID 35936508.
  5. Helgason, S (1959). "Differential operators on homogeneous spaces". Acta Mathematica. Institut Mittag-Leffler. 102 (3–4): 239–299. doi:10.1007/BF02564248.

References


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