New Examples of Irreducible Local Diffusion of Hyperbolic PDE's

TL;DR

This paper demonstrates that local diffusion phenomena in hyperbolic PDEs, previously thought reducible to simple singularities, can also occur at complex singularities, challenging existing assumptions.

Contribution

It provides the first examples of non-reducible local diffusion at complex singularities in generic wavefronts, expanding understanding of hyperbolic PDE behavior.

Findings

Diffusion occurs at non-simple singularities in hyperbolic PDEs.

Examples show diffusion is not always reducible to simple singular points.

Challenges previous conjectures about wavefront singularities.

Abstract

Local diffusion of strictly hyperbolic higher-order PDE's with constant coefficients at all simple singularities of corresponding wavefronts can be explained and recognized by only two local geometrical features of these wavefronts. We radically disprove the obvious conjecture extending this fact to arbitrary singularities: namely, we present examples of diffusion at all non-simple singularity classes of generic wavefronts in odd-dimensional spaces, which are not reducible to diffusion at simple singular points.