AIDJEX Bulletin #40 - Polar Science Center - University of Washington

differential equation.
2. Discontinuities of a solution cannot occur except along characteristics.
3. Characteristics are the only possible branch lines of solutions,
i.e., lines for which the same initial value problem may have several solutions.
The first property, which shall be exploited in the derivation of characteristic
directions and relations, is inherent in the basic fact: A direction
is characteristic at a point P if there exists a linear combination of the
differential equations for which all the unknowns are differentiated at P only
in this direction. A system of equations is said to be hyperbolic if it can
be replaced by a linearly equivalent one in which each differential equation
contains at every point differentiation in only one characteristic direction.
The second and third properties will be more useful in interpreting the
results of this analysis.
In this section we present formally the approach both for determining
the existence of characteristic directions and for determining the equations
that hold along each such curve.
Assume we are given the system of n equations in n anknowns:
which is chosen to have an appearance identical to the system of the four
equations shown in (27). If we pre-multiply by the vector RT to obtain
R T (43, + BZ + E) = 0,
.
--Y
we obtain one single scalar equation which is a linear combination of the
equations.
If there exist n distinct roots and n eigenvectors to the system
RL (Aa A, - E ) = 0
-a
a = 1, 2, ..., n
(33)
then for each such eigenvector R
'a
that
there is a linear combination of (31) so
124