22 1. Simple Examples of Propagation
There are conservations of all orders. Multiplying the spring energy
by ξ
2s
and integrating dξ shows that each of the following quantities is
independent of time:
1
2
∇t,xu(t)
2
Hs(Rd)
= ξ
2s |ξ|2
(
|a+(ξ)|2
+
|a−(ξ)|2
)
dξ .
Consider initial data a wave packet with wavelength of order and phase
equal to x1/,
(1.4.4) u (0,x) = γ(x)
eix1/
, ut(0,x) = 0 , γ ∈
s
Hs(Rd)
.
The choice ut = 0 postpones dealing with the factor 1/|ξ| in (1.4.3). When
is small, the initial value is an envelope or profile γ multiplied by a rapidly
oscillating exponential.
Applying (1.4.3) with g = 0 and
ˆ(ξ)
f = ˆ(0,ξ) u = F
(
γ(x)
eix1/
)
= ˆ(ξ γ − e1/)
yields u = u+ + u− with
u±(t, x) :=
1
2
1
(2π)d/2
ˆ(ξ γ − e1/)
ei(xξ∓t|ξ|)
dξ .
Analyze u+. The other term is analogous. For ease of reading, the
subscript plus is omitted. Introduce
ζ := ξ − e1/, so ξ =
e1 + ζ
, and
u (t, x) =
1
2
1
(2π)d/2
ˆ(ζ) γ
eix(e1+ζ)/ e−it|e1+ζ|/
dζ . (1.4.5)
The approximation of geometric optics comes from injecting the first order
Taylor approximation,
e1 + ζ ≈ 1 + ζ1 ,
yielding
uapprox :=
1
2
1
(2π)d/2
ˆ(ζ) γ
eix(e1+ζ)/ e−it(1+ζ1)/
dζ .
The rapidly oscillating terms
ei(x1−t)/
do not depend on ζ, so
(1.4.6)
uapprox =
ei(x1−t)/
A(t, x), A(t, x) :=
1
2
1
(2π)d/2
ˆ(ζ) γ
ei(xζ−tζ1)
dζ.
Write xζ − tζ1 = (x − te1)ζ to find
A(t, x) =
1
2
1
(2π)d/2
ˆ(ζ) γ
ei(x−te1)ζ
dζ =
1
2
γ(x − te1) .
