Singularities of dispersion relations

Abstract

SynopsisGeneric singularities occurring in dispersion relations are discussed within the framework of imperfect bifurcation theory and classified up to codimension four. Wave numbers are considered as bifurcation variables x =(x1,…, xn) and the frequency is regarded as a distinguished bifurcation parameter λ. The list of normal forms contains, as special cases, germs of the form ±λ +f(x), where f is a standard singularity in the sense of catastrophe theory. Since many dispersion relations are ℤ(2)-equivariant with respect to the frequency, bifurcation equations which are ℤ(2)-equivariant with respect to the bifurcation parameter are introduced and classified up to codimension four in order to describe generic singularities which occur at zero frequency. Physical implications of the theory are outlined.

The wave equation on the lattice in two and three dimensions

“…Some works, such as [2] and [3], have identified effects of spatial discretization on the PDE wave equation but with the goal of choosing discretization schemes that minimize these effects. The possible geometries of the singular surfaces are described using singularity theory in [1] and [4] without performing an analysis of the solution on the surfaces. The present work is apparently the first to find l pestimates of solutions to the lattice wave equation in the presence of these surfaces.…”

Remarks on the singularity indices of Arnol'd and Berry

“…For accounts of this theory and references to the literature see Guillemin and Sternberg [18], Duistermaat [15], Hdrmander [19], Arnol'd [1,3], and for applications to phenomena in statistical optics see Berry [6], Berry and Upstill [8]. Applications in other areas of physics are explored by Dangelmayr and co-workers [11][12][13][14] and by Berry [7].…”