Representation of solutions to wave equations with profile functions

Abstract

Solutions to the wave equation with constant coefficients in R d can be represented explicitly in Fourier space. We investigate a reconstruction formula, which provides an approximation of solutions u(., t) to initial data u 0 (.) for large times. The reconstruction consists of three steps: 1) Given u 0 , initial data for a profile equation are extracted.2) A profile evolution equation determines the shape of the profile at time τ = ε 2 t. 3) A shell reconstruction operator transforms the profile to a function on R d . The sketched construction simplifies the wave equation, since only a one-dimensional problem in an O(1) time span has to be solved. We prove that the construction provides a good approximation to the wave evolution operator for times t of order ε −2 .