The author constructs the theory of reticular Legendrian unfoldings that describes stable and generic bifurcations of wavefronts generated by a hypersurface germ with a boundary, a corner, or an rcorner (cf. [4]).
In the case
, the hypersurface has no boundary; this is known as the theory of perestroikas of fronts ([1], the figures are given on p. 60). A oneparameter 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 bifurcations 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 I, Basel: Birkhäuser, 1985.
[3] T. Tsukada, "Genericity of Caustics and Wavefronts on an rCorner",
14(3),
The Asian Journal of Mathematics, 2010 pp. 335–358.
[4] T. Tsukada, "Bifurcations of Wavefronts on rCorners: SemiLocal Classification,"
18(3)
Methods and Applications of Analysis, 2011 pp. 303–334.