owe a great intellectual debt to the lecture notes, and to all that Peter Lax
has taught me through the years.
The book introduces a large and rich subject and I hope that readers
are suﬃciently attracted to probe further.
P.2. A bird’s eye view of hyperbolic equations
The central theme of this book is hyperbolic partial differential equations.
These equations are important for a variety of reasons that we sketch here
and that recur in many different guises throughout the book.
The first encounter with hyperbolicity is usually in considering scalar
real linear second order partial differential operators in two variables with
coeﬃcients that may depend on x,
a ux1x1 + b ux1x2 + c ux2x2 + lower order terms .
Associate the quadratic form ξ → a ξ1
+ b ξ1ξ2 +c ξ2.
The differential opera-
tor is elliptic when the form is positive or negative definite. The differential
operator is strictly hyperbolic when the form is indefinite and nondegener-
In the elliptic case one has strong local regularity theorems and solv-
ability of the Dirichlet problem on small discs. In the hyperbolic cases, the
initial value problem is locally well set with data given at noncharacteristic
curves and there is finite speed of propagation. Singularities or oscillations
in Cauchy data propagate along characteristic curves.
The defining properties of hyperbolic problems include well posed
Cauchy problems, finite speed of propagation, and the existence of wave
like structures with infinitely varied form. To see the latter, consider in
initial data on t = 0 with the form of a short wavelength wave packet,
localized near a point p. The solution will launch wave packets
along each of two characteristic curves. The envelopes are computed from
those of the initial data, as in §5.2, and can take any form. One can send
essentially arbitrary amplitude modulated signals.
The infinite variety of wave forms makes hyperbolic equations the pre-
ferred mode for communicating information, for example in hearing, sight,
television, and radio. The model equations for the first are the linearized
compressible inviscid fluid dynamics, a.k.a. acoustics. For the latter three it
is Maxwell’s equations. The telecommunication examples have the property
that there is propagation with very small losses over large distances. The
examples of wave packets and long distances show the importance of short
wavelength and large time asymptotic analyses.
form is nondegenerate when its defining symmetric matrix is invertible.