Typical Bifurcations of Wavefront Intersections

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This Demonstration shows all generic bifurcations of intersections of wavefronts generated by a hypersurface with or without a boundary in a smooth -dimensional manifold for , . The time can be varied with a slider.

Contributed by: Takaharu Tsukada (April 2012)
Open content licensed under CC BY-NC-SA


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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 r‐corner (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 one-parameter 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 multi-reticular Legendrian unfoldings in order to construct a generic classification of semi-local situations.

A multi-reticular Legendrian unfolding consists of products of reticular Legendrian unfoldings. Its wavefronts are unions of wavefronts of the reticular Legendrian unfoldings.

A multi-generating family of a generic multi-reticular Legendrian unfolding () is reticular ‐equivalent to one of the following:

.

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," Asian Journal of Mathematics, 14(3), 2010 pp. 335–358.

[4] T. Tsukada, "Bifurcations of Wavefronts on r-Corners: Semi-Local Classifications," Methods and Applications of Analysis, 18(3), 2011, pp. 303–334.



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