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 =

∂xv

3

(in the sense of distributions) since they both satisfy the same initial

value problem. Thus v and u = ∂xv

3

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

ei(τt−xξ).

Discussion. The solutions

ei(τt−xξ)

with

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

S(Rd)

the Schwartz

space of rapidly decreasing smooth functions. That is, functions such that

for all α, β,

sup

x∈Rd

xβ∂x αψ(x)

∞ .

Exercise 1.2.3. Prove that if u ∈

C∞(R

: S(R)) is a real valued solution

of the Klein–Gordon equation, then

ut

2

+ ux

2

+

u2

dx

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

⎛

⎝0

1 0 −1

1

−1⎠

0 0 −1

⎞

V .