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/ , u t (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) = ˆ(0,ξ) = F ( γ(x) eix1/ ) = ˆ(ξ − e1/) yields u = u+ + u− with u ± (t, x) := 1 2 1 (2π)d/2 ˆ(ξ − 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) 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 u approx := 1 2 1 (2π)d/2 ˆ(ζ) 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) ei(xζ−tζ1) dζ. Write xζ − tζ1 = (x − te1)ζ to find A(t, x) = 1 2 1 (2π)d/2 ˆ(ζ) ei(x−te1)ζ dζ = 1 2 γ(x − te1) .
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