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)
, 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
)
.
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.
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