# Multi-Time Bifurcations of Wavefronts in 2D and 3D

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This Demonstration shows all "generic" multi-bifurcations of wavefronts in 2D and 3D. You can control the time.

Contributed by: Takaharu Tsukada (October 2013)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

In [5], the author constructs the theory of reticular Legendrian unfoldings that describes stable and generic multi-bifurcations of wavefronts generated by a hypersurface germ with a boundary, a corner, or an ‐corner.

In the case , the hypersurface has no boundary; a two-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 multi-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 :

, , , .

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, Volume I: The Classification of Critical Points, Caustics and Wave Fronts* (*Monographs in Mathematics*, Vol. 82), Basel: Birkhäuser, 1985.

[3] T. Tsukada, "Genericity of Caustics and Wavefronts on an -Corner," *Asian Journal of Mathematics*, 14(3), 2010 pp. 335–358. projecteuclid.org/euclid.ajm/1295040754.

[4] T. Tsukada, "Bifurcations of Wavefronts on -Corners: Semi-Local Classification," *Methods and Applications of Analysis*, 18(3), 2011 pp. 303–334. doi:10.4310/MAA.2011.v18.n3.a3.

[5] T. Tsukada. "Multi-Bifurcations of Wavefronts on -Corners." arxiv.org/abs/1308.2274.

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