The Ultrahyperbolic Differential Equation with Four Independent Variables

Abstract

The properties of a linear homogeneous partial differential equation of the second order

$$\sum\limits_{i,k} {{a_{ik}}} \frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_k}}} + \sum\limits_i {{b_i}} \frac{{\partial u}}{{\partial {x_i}}} + cu = 0$$

((1))

for a function, u(x ₁, • • •, x _n) are known to depend largely on the index of the quadratic form Q(ξ) =

$$\sum\limits_{i,k} {{a_{ik}}{\xi _i}} {\xi _k}$$

.¹ If by a suitable real linear transformation Q can be brought into the form ±(ξ ²₁ +• • • + ξ ² _n , i.e., if Q is definite, (1) is called an elliptic equation. If Q can be transformed into ±(ξ ²₁ + • • • + ξ ² _n-1 - ξ ² _n ), the equation is called normal hyperbolic. Elliptic and normal hyperbolic equations constitute the two types which have been studied more extensively, besides the case of a parabolic equation for which det (a _ik ) = 0. Equations which are neither elliptic, nor parabolic, nor normal hyperbolic, i.e., equations for which the corresponding quadratic form Q can be written in the form ξ ²₁ + ξ ²₂ ± • • • ± ξ ² _n-2 - ξ ² _n-1 - ξ ² _n , have scarcely been treated, at least not without restriction to solutions which are analytic in all or some of the variables. For such equations the notation ultrahyperbolic has been introduced by R. Courant.