Algebraic Conditions for Hyperbolicity of Systems of Partial Differential Equations

Abstract

Partial differential equations or systems of such equations are classified according to “type”, such as “elliptic”, “parabolic”, “hyperbolic”, etc. Hyperbolic systems (with respect to the initial plane t = 0) are those for which the initial value problem is well posed in the sense of Hadamard (see [8]). General conditions that are both necessary and sufficient for hyperbolicity are complicated (see [10], [11], [2], [3], [14], [15], [16], [17]). The situation, however, is rather simple for an m-th order linear homogeneous system of equations with constant coefficients:

$$ P\left( {\frac{\partial }{{\partial t}},\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}} \right)u = 0 $$

((1))

for a vector u = u(t, x ₁ • • •, x _n ) = u(t, x) with N components. Here P is an N x N square matrix whose elements p _ik are m-th degree forms in their n + 1 arguments. We associate with (1) the matrix (the “symbol” of (1))

$$ P(\lambda ,\xi ) = P(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) = (Pik(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n})) $$

((2))

and the characteristic form

$$ Q(\lambda ,\xi ) = Q(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) = \det P(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) $$

((3))

of degree mN in its arguments.

The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014–76-C-0301. Reproduction in whole or in part is permitted for any purpose of the United States Government. A summary of results of this paper has appeared in [12].