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On the local stability of wave fronts defined by eikonal equations

Published online by Cambridge University Press:  14 November 2011

David Rand
David Rand
Affiliation:
Mathematics Institute, University of Warwick, Coventry
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Synopsis

Associated with a Cauchy problem for an eikonal equation is a structure of caustics and wave fronts. The geometrical properties of these structures are of considerable interest and one would like to classify them. The first step in such a programme is to identify those solutions whose structures have certain (structural) stability properties or robustness. These are structures which survive small perturbations of the initial conditions. (It turns out that they also survive small perturbations of the equation.) This problem for caustics has been tackled with considerable success by a number of authors, in particular by Arnold. Here we develop the analogous, but different, theory for wave fronts and give an algebraic characterization of those local solutions of eikonal equations which have stable wave front structures. From this result it is easy to deduce the qualitative form of those stable wave front structures which occur when the propagation occurs in a space of dimension not greater than five. They correspond to Thorn's elementary catastrophes, as demonstrated by Zakalyukin.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 85 , Issue 3-4 , 1980 , pp. 195 - 232
Copyright
Copyright © Royal Society of Edinburgh 1980

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