16 1. Simple Examples of Propagation

along codimension one characteristic hypersurfaces in space-time. The sin-

gularities propagate satisfying transport equations along rays generating the

hypersurface. The general class goes under the name conormal solutions.

M. Beals’ book [Beals, 1989] is a good reference. They describe propagating

wavefronts. Luneberg’s book [Luneberg, 1944] recounts his discovery that

the propagation laws for fronts of singularities coincide with the physical

laws of geometric optics.

1.3. Group velocity and the method of nonstationary phase

The Klein–Gordon equation has constant coeﬃcients, and so it can be solved

explicitly using the Fourier transform. The computation of the singularities

of the fundamental solution of the Klein–Gordon equation in Exercise 1.2.5

suggests that the main part of solutions travel with speed equal to 1. One

might expect that the energy in a disk growing linearly in time at a speed

1 would be small for t 1. For compactly supported data, such a disk

would contain no singularities for large time. Thus it is not unreasonable to

guess that for each σ 1 and R 0,

(1.3.1) lim sup

t→∞

|x|R+σt

ut

2

+ ux

2

+

u2

dx = 0.

Either the method of characteristics or the energy method shows that

speeds are no larger than one. The idea about the Klein–Gordon energy

expressed in (1.3.1) is dead wrong. The main part of the energy travels

strictly slower than speed 1, even though singularities travel with speed

exactly equal to 1.

The solution of the Cauchy problem for the Klein–Gordon equation in

dimension d,

utt − Δ u + u = 0 , (t, x) ∈

R1+d

,

is given by

u =

±

(2π)−d/2

a±(ξ)

ei(±ξ t+xξ)

dξ , ξ :=

(

1 +

|ξ|2

)1/2

,

ˆ(0,ξ) u = a+(ξ) + a−(ξ) , ˆt(0,ξ) u = i ξ

(

a+(ξ) − a−(ξ)

)

.

The conserved energy is equal to

1

2

ut

2

+

|∇xu|2

+

u2

dx = ξ

2

(

|a+(ξ)|2

+

|a−(ξ)|2

)

dξ .

Exercise 1.3.1. Verify these formulas. Verify conservation of energy by

an integration by parts argument as in Exercise 1.2.3. Hint. Follow the

computation that starts §1.4.