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In the [[mathematics|mathematical]] field of [[partial differential equation]]s, the '''ultrahyperbolic  equation''' is a partial differential equation for an unknown scalar function ''u'' of 2''n'' variables ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, ''y''<sub>1</sub>, ..., ''y''<sub>''n''</sub> of the form

: <math>\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.\qquad\qquad(1)</math>

More generally, if ''a'' is any [[quadratic form]] in 2''n'' variables with [[metric signature|signature]] (''n'',''n''), then any PDE whose principal part is <math>a_{ij}u_{x_ix_j}</math> is said to be ultrahyperbolic.  Any such equation can be put in the form 1. above by means of a change of variables.<ref name="See Courant and Hilbert">See Courant and Hilbert.</ref>

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 [[Method of characteristics|characteristics]], one of which is due to [[Fritz John]]: the [[John equation]].

Walter Craig and Steven Weinstein recently (2008) proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one hypersurface.<ref>{{cite web | url=http://rspa.royalsocietypublishing.org/content/465/2110/3023 | title = On determinism and well-posedness in multiple time dimensions | publisher = Proc. R. Soc. A  vol. 465 no. 2110 3023-3046 (2008) | last1= Craig | first1 = Walter | last2 = Weinstein | first2 = Steven | accessdate = 5 December 2013}}</ref>

The equation has also been studied from the point of view of [[symmetric space]]s, and [[elliptic differential operator]]s.<ref>See, for instance, Helgasson.</ref>  In particular, the ultrahyperbolic  equation satisfies an analog of the [[harmonic function|mean value theorem for harmonic functions]]

==Notes==
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==References==
* {{cite book|title=Methods of Mathematical Physics, Vol. 2|author=[[David Hilbert]] and [[Richard Courant]]|publisher=Wiley-Interscience|isbn=978-0-471-50439-9|pages=744–752|year=1962}}
* {{cite journal|author=[[Lars Hörmander]]|title=Asgeirsson's Mean Value Theorem and Related Identities|journal=Journal of Functional Analysis|year=2001|issue=184|pages=377–401}}
* {{cite book|author=Lars Hörmander|title=The Analysis of Linear Partial Differential Operators I|publisher=Springer-Verlag|year=1990|isbn=3-540-52343-X|pages=Theorem 7.3.4}}
* {{cite book|title=Groups and Geometric Analysis|author=Sigurdur Helgason|year=2000|publisher=American Mathematical Society|pages=319–323}}
* {{cite journal|author=[[Fritz John]]|title=The Ultrahyperbolic Differential Equation with Four Independent Variables|journal=Duke Math. J.|volume=4|issue=2|year=1938|pages=300–322|doi=10.1215/S0012-7094-38-00423-5}}

[[Category:Differential operators]]


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