the ultrahyperbolic type
The familiar trio — elliptic, parabolic, hyperbolic — is the complete story only in TWO independent variables, where one number, the discriminant, sorts everything. In three or more variables a fourth possibility quietly appears, one with no two-variable analogue: the ultrahyperbolic type. It is the type that arises when an equation behaves like a wave in more than one time-like direction at once.
To classify in many variables you look at the matrix of second-order coefficients and count the signs of its eigenvalues (its signature). Elliptic means all eigenvalues have the same sign (the Laplacian u_xx + u_yy + u_zz + u_ww, all plus). Hyperbolic means exactly one sign is the odd one out — one minus and the rest plus, as in the wave operator u_tt - u_xx - u_yy - u_zz, with a single time-like direction. Ultrahyperbolic is what remains: TWO OR MORE eigenvalues of each sign, with none zero — for example an operator like u_tt + u_ss - u_xx - u_yy in four variables, which has two plus and two minus signs. There is no such creature with only two variables, because two eigenvalues can give at most one of each sign (that is just hyperbolic).
Ultrahyperbolic equations are mathematical curiosities more than everyday workhorses, but they are genuinely instructive. The classic fact (Asgeirsson's mean-value theorem) is that the naive initial-value problem for them is generally ILL-POSED — you cannot freely prescribe Cauchy data and expect a solution to exist and depend continuously on it, which is a vivid reminder that well-posedness is tied to type. They do appear in real settings — for instance in integral geometry and tomography, where the X-ray / John transform satisfies an ultrahyperbolic equation. The lesson to carry away is that the three-way classification is a two-variable convenience; the honest, general statement classifies by the symbol's signature, and ultrahyperbolic is the case the two-variable picture cannot see.
In four variables the operator u_{t1 t1} + u_{t2 t2} - u_{x1 x1} - u_{x2 x2} = 0 is ultrahyperbolic: its coefficient matrix has eigenvalues +1, +1, -1, -1 — two of each sign — so it is neither elliptic (not all the same), nor hyperbolic (not just one odd sign), nor parabolic (none are zero).
Ultrahyperbolic needs at least four variables: two or more of each sign, none zero — the case the elliptic/parabolic/hyperbolic trio cannot reach.
Ultrahyperbolic equations are the standard counterexample to the lazy belief that every PDE is elliptic, parabolic, or hyperbolic — that trichotomy is exhaustive only in two variables, and the naive Cauchy problem for an ultrahyperbolic equation is typically ill-posed.