In [5], the author constructs the theory of reticular Legendrian unfoldings that describes stable and generic multibifurcations of wavefronts generated by a hypersurface germ with a boundary, a corner, or an
corner.
In the case
, the hypersurface has no boundary; a twoparameter family of wavefronts
is given by a generating family
defined on
such that
.
In the case
, the hypersurface has a boundary; a reticular Legendrian unfolding gives the wavefront
, where the set
is the wavefront generated by the hypersurface at time
, and the set
is the wavefront generated by the boundary of the hypersurface at time
.
A reticular Legendrian unfolding has a generating family. Then the wavefront
is given by the generating family
defined on
such that
,
.
Typical multibifurcations of wavefronts in 2D and 3D are defined by generic reticular Legendrian unfoldings for the cases
. Their generating families are stably reticular


equivalent to one of the following:
In the case
,
:
In the case
:
[1] V. I. Arnold,
Singularities of Caustics and Wave Fronts, Dordrecht: Kluwer Academic Publishers, 1990.
[2] V. I. Arnold, S. M. Gusein–Zade, and A. N. Varchenko,
Singularities of Differential Maps, Volume I: The Classification of Critical Points, Caustics and Wave Fronts (
Monographs in Mathematics, Vol. 82), Basel: Birkhäuser, 1985.
[4] T. Tsukada, "Bifurcations of Wavefronts on
Corners: SemiLocal Classification,"
Methods and Applications of Analysis,
18(3), 2011 pp. 303–334.
doi:10.4310/MAA.2011.v18.n3.a3.