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) .