14 1. Simple Examples of Propagation
The Cauchy data, V (0,x) are continuous, piecewise smooth, and singular
only at x = 0. Theorem 1.1.4 shows that V is piecewise smooth with
singularities only on the characteristics through (0, 0). In addition u =
(in the sense of distributions) since they both satisfy the same initial
value problem. Thus v and u = ∂xv
have singular support only on the
characteristics through (0, 0).
Interesting things happen if one adds a lower order term. For example,
consider the Klein–Gordon equation
(1.2.6) utt uxx + u = 0 .
In sharp contrast with equation (1.2.2), there are hardly any undistorted
progressing wave solutions.
Exercise 1.2.2. Find all solutions of (1.2.6) of the form f(x ct) and all
solutions of the form
Discussion. The solutions
ξ R are particularly important since the general solution is a Fourier
superposition of these special plane waves. The equation τ = τ(ξ) defining
such solutions is called the dispersion relation of (3.1.6).
There is an energy conservation law. Denote by
the Schwartz
space of rapidly decreasing smooth functions. That is, functions such that
for all α, β,
xβ∂x αψ(x)
Exercise 1.2.3. Prove that if u
: S(R)) is a real valued solution
of the Klein–Gordon equation, then
+ ux
is independent of t. This quantity is called the energy. Hint. Justify
carefully differentiation under the integral sign and integration by parts. If
you find weaker hypotheses which suffice, that is good.
The fundamental solution of the Klein–Gordon equation with initial data
(1.2.4), is not as simple as the fundamental solutions of the wave equation.
Theorem 1.1.4 implies that the singular support lies on {x = ±t}. The proof
is as for the wave equation except that the zeroth order term in the equation
for V is replaced by

1 0 −1
0 0 −1

V .
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