Typical Bifurcations of Wavefronts in 2D and 3D

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 r-corner (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 one-parameter 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 :

The wavefronts for singularities are given in the Demonstration Typical Wavefronts in 2D and 3D.

References

[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 r-Corner", publication forthcoming in Asian J. of Math., 2010.

[4] T. Tsukada, "Bifurcations of Wavefronts on an r-Corner," arXiv, 2009.

[5] T. Tsukada, "Bifurcations of Wavefronts on an r-Corner II: Semi-Local Classification," arXiv, 2009.