In this Demonstration, stable reticular Legendrian unfoldings and generic bifurcations of wavefronts are generated by a hypersurface germ with a boundary, a corner, or an rcorner (cf. [4]).
For the case
, the hypersurface has no boundary; the fronts are described as perestroikas (in [1] the figures are given on p. 60). A oneparameter family of wavefronts
is given by a generating family
defined on
such that
.
For 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.
For
:
For
:
Typical wavefronts in 2D and 3D are shown for
singularities while typical bifurcations in 2D and 3D are shown for
singularities.
The author also applies the theory of multireticular Legendrian unfoldings in order to construct a generic classification of semilocal situations.
A multireticular Legendrian unfolding consists of
products of reticular Legendrian unfoldings. Its wavefronts are unions of wavefronts of the reticular Legendrian unfoldings.
A multigenerating family of a generic multireticular Legendrian unfolding (
) is reticular


equivalent to one of the following:
.
In this Demonstration all generic bifurcations of
intersections are given for wavefronts in an
dimensional manifold for
,
.
[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,"
Asian Journal of Mathematics,
14(3), 2010 pp. 335–358.
[4] T. Tsukada, "Bifurcations of Wavefronts on rCorners: SemiLocal Classifications,"
Methods and Applications of Analysis,
18(3), 2011, pp. 303–334.