Representation of solutions to wave equations with profile functions

Abstract

In many applications, solutions to wave equations can be represented in Fourier space with the help of a dispersion function. Examples include wave equations on periodic lattices with spacing [Formula: see text], wave equations on [Formula: see text] with constant coefficients, and wave equations on [Formula: see text] with coefficients of periodicity [Formula: see text]. We characterize such solutions for large times [Formula: see text]. We establish a reconstruction formula that yields approximations for solutions in three steps: (1) From given initial data [Formula: see text], appropriate initial data for a profile equation are extracted. (2) The dispersion function determines a profile evolution equation, which, in turn, yields the shape of the profile at time [Formula: see text]. (3) A shell reconstruction operator transforms the profile to a function on [Formula: see text]. The resulting function is a good approximation of the solution [Formula: see text].