Bifurcations of Wavefronts on an r-corner

TL;DR

This paper introduces reticular Legendrian unfoldings to analyze the stability and classification of wavefront bifurcations generated by hypersurfaces with boundaries or corners in low-dimensional manifolds.

Contribution

It defines new stability concepts for reticular Legendrian unfoldings and establishes their equivalence with generating family stabilities, providing a classification of generic bifurcations.

Findings

Classification of generic bifurcations for r=0, n<=5

Classification of generic bifurcations for r=1, n<=3

Equivalence of stabilities of unfoldings and generating families

Abstract

We introduce the notion of reticular Legendrian unfoldings in order to investigate stabilities of bifurcations of wavefronts generated by a hypersurface germ with a boundary, a corner, or an r-corner in a smooth n dimensional manifold. We define several stabilities of reticular Legendrian unfoldings and prove that they and the stabilities of corresponding generating families are all equivalent and give the classification of generic bifurcations of wavefronts in the cases r=0,n<=5 and r=1,n<= 3respectively.