Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-01T02:08:48.750Z Has data issue: false hasContentIssue false
  • English
  • Français

Singularities of dispersion relations

Published online by Cambridge University Press:  14 November 2011

G. Dangelmayr
G. Dangelmayr
Affiliation:
Institute for Information Sciences, University of Tübingen, Tübingen, Federal Republic of Germany
Get access Rights & Permissions [Opens in a new window]

Synopsis

Generic singularities occurring in dispersion relations are discussed within the framework of imperfect bifurcation theory and classified up to codimension four. Wave numbers are considered as bifurcation variables x =(x1,…, xn) and the frequency is regarded as a distinguished bifurcation parameter λ. The list of normal forms contains, as special cases, germs of the form ±λ +f(x), where f is a standard singularity in the sense of catastrophe theory. Since many dispersion relations are ℤ(2)-equivariant with respect to the frequency, bifurcation equations which are ℤ(2)-equivariant with respect to the bifurcation parameter are introduced and classified up to codimension four in order to describe generic singularities which occur at zero frequency. Physical implications of the theory are outlined.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 95 , Issue 3-4 , 1983 , pp. 301 - 316
Copyright
Copyright © Royal Society of Edinburgh 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Arnold, V.. Critical points of smooth functions. In Proceedings of the International Congress of Mathematicians, Vol. I, Vancouver, 1974 (Montreal: Canad. Math. Congress, 1975).Google Scholar
2Berry, M. V.. Waves and Thorn's theorem. Adv. Phys. 25 (1976), 1–26.Google Scholar
3Berry, M. V. and Upstill, C.. Catastophe optics: morphologies of caustics and their diffraction patterns. In: Wolf, E. (ed.), Progress in Optics 18 (1980), 259–346.Google Scholar
4Dangelmayr, G.. Analytic singularities in dispersive wave phenomena based on topological singularities of dispersion relations. J. Phys. A 15 (1982), L337–L342.Google Scholar
5Dangelmayr, G. and Güttinger, W.. Topological approach to remote sensing. Geophys. J. Roy. Astr. Soc. 71 (1982), 79–126.Google Scholar
6Gibson, C. G., Singular points of smooth mappings (London: Pitman, 1979).Google Scholar
7Golubitsky, M. and Schaeffer, D.. A theory for imperfect bifurcations via singularity theory. Comm. Pure Appl. Math. 32 (1979), 21–98.CrossRefGoogle Scholar
8Golubitsky, M. and Schaeffer, D.. Imperfect bifurcations in the presence of symmetry. Comm. Math. Phys. 67 (1979), 205–232.Google Scholar
9Golubitsky, M. and Keyfitz, B. L.. A qualitative study of the steady-state solutions for a continuously stirred tank chemical reactor. SIAM J. Math. Anal. 11 (1980), 316–339.Google Scholar
10Golubitsky, M., Keyfitz, B. L. and Schaeffer, D.. A singularity theory analysis of a thermalchainbranching model for the explosion peninsula. Comm. Pure Appl. Math. 34 (1981), 433–463.Google Scholar
11Lighthill, J.. Waves in fluids (Cambridge Univ. Press, 1979).Google Scholar
12Poston, T.. Perturbed bifurcations in crystal spectra, preprint. Geneva, 1979.Google Scholar
13Poston, T. and Stewart, I.. Catastrophe theory and its applications (London: Pitman, 1978).Google Scholar
14Schwartz, G.. Smooth functions invariant under the action of a compact Lie group. Topology 14 (1975), 63–68.Google Scholar
15Siersma, D.. Classification and deformation of singularities (Thesis, University of Utrecht, 1974).Google Scholar
16Stewart, I.. Applications of catastrophe theory to the physical sciences. Physica 2D (1981), 245–305.Google Scholar
17Thorn, R.. Structural stability and morphogenesis (New York: Benjamin, 1975).Google Scholar
18Hove, L. Van. The occurrence of singularities in the elastic frequency distribution of a crystal. Phys. Rev. 89 (1953), 1189–1193.Google Scholar
19Wannier, G. H.. Elements of solid state theory (Cambridge Univ. Press, 1959).Google Scholar
20Wassermann, G.. Stability of unfoldings. Lecture Notes in Mathematics 393 (Berlin: Springer, 1974)Google Scholar