Propagation estimates for dispersive wave equations: Application to the stratified wave equation

Abstract

The plane-stratified wave equation (∂t2+H)ψ=0 with H=−c(y)2∇z2 is studied, where z=x⊕y, x∈Rk, y∈R1 and |c(y)−c∞|→0 as |y|→∞. Solutions to such an equation are solved for the propagation of waves through a layered medium and can include waves which propagate in the x-directions only (i.e., trapped modes). This leads to a consideration of the pseudo-differential wave equation (∂t2+ω(−Δx))ψ=0 such that the dispersion relation ω(ξ2) is analytic and satisfies c1⩽ω′(ξ2)⩽c2 for c*>0. Uniform propagation estimates like ∫|x|⩽|t|αE(UtP±φ0)dkx⩽Cα,β(1+|t|)−β∫E(φ0)dkx are obtained where Ut is the evolution group, P± are projection operators onto the Hilbert space of initial conditions φ∈H and E(⋅) is the local energy density. In special cases scattering of trapped modes off a local perturbation satisfies the causality estimate ‖P+ρΛjSP−ρΛk‖⩽Cνρ−ν for each ν<1/2. Here P+ρΛj (P−ρΛk) are remote outgoing/detector (incoming/transmitter) projections for the jth (kth) trapped mode. Also Λ⋐R+ is compact, so the projections localize onto formally-incoming (eventually-outgoing) states.