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Caustics, Catastrophes and Wave Fields pp 48–72Cite as

Typical Integrals of Catastrophe Theory

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Part of the book series: Springer Series on Wave Phenomena ((SSWAV,volume 15))

Abstract

This chapter is devoted to standard integrals associated with typical caustics. We present the simplest integrals corresponding to the seven Thom’s catastrophes and also some more complex integrals associated with caustics that occur in series. A novel point in this exposition is the one of Fresnel’s criteria for passing over from some normal forms to others when external parameters vary.

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References

  1. V.I. Arnold, A.N. Varchenko, S.M. Gussein-Zade. Singularities of Dijferentiable Mappings, Vols. 1, 2 (Birkhäuser, Boston 1985, 1988)

    Book  Google Scholar 

  2. J.J Dustermaat: Commun. Pure Appl. Math. 27, 207 (1974)

    Article  Google Scholar 

  3. M.V. Berry: J. Phys. A 10, 2061 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  4. M.V. Berry, C. Upstill: Progress in Optics 18, 257–320 (North-Holland, Amsterdam 1980)

    Google Scholar 

  5. Yu.A. Kravtsov, Yu.I. Orlov: Geometrical Optics of Inhomogeneous Media, Springer Ser. Wave Phenom., Vol. 6 (Springer, Berlin, Heidelberg 1990)

    Google Scholar 

  6. Yu.A. Kravtsov, Yu.I. Orlov: Sov. Phys. - Usp. 26, 1038 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  7. Yu.A. Kravtsov: Rays and caustics as physical objects. Progress in Optics 26, 227–348 (North-Holland, Amsterdam 1988)

    Google Scholar 

  8. V.I. Arnold: Funkts. Anal. Prilozh. 6, 61 (1972)

    Google Scholar 

  9. M.V. Fedoryuk: Metod Perevala (saddle-point method) (Nauka, Moscow 1977)

    MATH  Google Scholar 

  10. R.B. Dingle: Asymptotic Expansions: Their Derivation and Interpretation (Academic, London 1973)

    MATH  Google Scholar 

  11. A.A. Asatryan, Yu.A. Kravtsov: Wave Motion 10, 45 (1988)

    Article  Google Scholar 

  12. J.C.P. Miller: The Airy Integral Giving Tables of Solution of the Differential Equation y″ = xy (Cambridge Univ. Press, Cambridge 1971)

    Google Scholar 

  13. V.A. Fock: Electromagnetic Diffraction and Propagation Problems (Pergamon, New York 1965)

    Google Scholar 

  14. M. Abramovitz, J.A. Stegun: Handbook of Mathematical Functions (NBS, Washington, DC 1964)

    Google Scholar 

  15. R.E. Langer: Trans. Am. Math. Soc. 31, 23 (1931)

    Article  Google Scholar 

  16. T.M. Cherry: Trans. Am. Math. Soc. 68, 224 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  17. A.A. Dorodnitsyn: Usp. Mat. Nauk 7, 3 (1952)

    Google Scholar 

  18. V.Ya. Groshev, Yu.A. Kravtsov: Izv. VUZ Radiofiz. 11, 1812 (1968) [English transl.: Radiophys. Quantum Electron. 11, 1019 (1968)]

    Google Scholar 

  19. T. Pearsey: Philos. Mag. 37, 311 (1946)

    MathSciNet  Google Scholar 

  20. L.M. Brekhovskikh, O.A. Godin: Acoustics of Layered Media I, II, Springer Ser. Wave Phenom., Vols. 5, 10 (Springer, Berlin, Heidelberg 1990, 1992)

    Google Scholar 

  21. A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Spetsialnye Funktsii Volnovykh Katastrof (special functions of wave catastrophes). Institute of Radio Engineering and Electronics AN SSSR, preprint No. 43 (415) (IRE AN SSSR, Moscow 1984)

    Google Scholar 

  22. J.N.L. Connor, D. Farrelly: J. Chem. Phys. 75, 2831 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  23. J.N.L. Connor, P.R. Curtis: J.. Phys. A 15, 1179 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Yu.A. Kravtsov, Yu.A. Orlov: Sov. Phys. - Usp. 23, 750 (1980); Radio. Sci. 16, 975 (1981)

    Google Scholar 

  25. D. Ludwig: Commun. Pure Appl. Math. 19, 215 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Trinkaus, F. Drepper: J. Phys. A 10, L11 (1977)

    Article  ADS  Google Scholar 

  27. N.F. Dronov, E.B. Ipatov, D.S. Lukin, E.A. Palkin: In Rasprostranenie Radiovoln v Ionosfere (radiowave propagation in the ionosphere) (ISMIRAN, Moscow 1978) p. 57

    Google Scholar 

  28. M.V. Berry, J.F. Nye, F.J. Wright: Philos. Trans. Roy. Soc. London 291, 453 (1979)

    Article  ADS  Google Scholar 

  29. D.S. Lukin, E.A. Palkin: Chislennyi KanonicheskiiMetod v Zadachakh Difraktsii i Rasprostranenia Elektromcignitnykh Voln v Neodnorodnykh Sredakh (numerical canonic method in problems of diffraction and propagation of electromagnetic waves in inhomogeneous media) (MFTI, Moscow 1982) p. 159.

    Google Scholar 

  30. J.N.L. Connor, P.R. Curtis, D. Farrelly: J. Phys. A 17, 283 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  31. V.I. Arnold: Usp. Mat. Nauk 28, 17 (1973)

    Google Scholar 

  32. J.N.L. Connor: Mol. Phys. 26, 1371 (1973)

    Article  ADS  Google Scholar 

  33. J.N.L. Connor: Faraday Discuss. Chem. Soc. 55, 51 (1973)

    Article  Google Scholar 

  34. J.N.L. Connor, D. Farrelly: Chem. Phys. Lett. 81, 306 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  35. R. Gilmore: Catastrophe Theory for Scientists and Engineers (Wiley, New York 1981)

    MATH  Google Scholar 

  36. D.S. Lukin, E.B.Ipatov, E.A. Palkin: In Voprosy difraktsii elektromagnitnykh voln (problems of electromagnetic waves diffraction) (MFTI, Moscow 1982) p. 21

    Google Scholar 

  37. E.B. Ipatov, D.S. Lukin, E.A. Palkin: Zhurn. Vychislit. Matematiki i Matematich. Fiziki 25, 224 (1985)

    MathSciNet  ADS  MATH  Google Scholar 

  38. E.B. Ipatov, A.S. Kryukovskii, D.S. Lukin, D. V. Rastyagaev: In The 15th Triennal International Symposium on Electromagnetic Theory (URSI, St. Petersburg 1985) p. 419

    Google Scholar 

  39. M.V. Berry: Adv. Phys. 25, 1 (1976)

    Article  ADS  Google Scholar 

  40. J.N.L. Connor; P.R. Curtis, D. Farrelly: Mol. Phys. 48, 1305 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  41. A.S. Kryukovskii, D.S. Lukin, E.A. Palkin: Kraevye i Uglovye Katastofy v Zadazhakh Difraktsii i Rasprostranenia Voln (the edge and corner catstrophes in problems of wave diffraction and propagation) (Kazanskii Aviatsionnyi Institut, Kazan 1988)

    Google Scholar 

  42. A.S. Kryukovskii: In Difraktsia i Rasprostranenie Electromagnitnykh i Akusticheskikh Voln (diffraction and propagation of electromagnetic and acoustics waves) (MFTI, Moscow 1992, p. 29)

    Google Scholar 

  43. A.S. Kryukovskii, D. S. Lukin, E.A. Palkin: Doklady RAN 341, 456 (1995)

    MathSciNet  Google Scholar 

  44. A.S. Kryukovskii, D. V. Rastyagaev: In Problemy rasprostranenia i Difraktsii Electromagnitnykh Voln (problems of electromagnetic waves propagation and diffraction) (MFTI, Moscow 1995) p. 54

    Google Scholar 

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Authors and Affiliations

  1. Space Research Institute, Russian Academy of Science, Profsoyyznaya 84/32, 117810, Moscow, Russia

    Yu. A. Kravtsov

  2. Power Engineering Institute, Moscow, Russia

    Yu. I. Orlov

Authors
  1. Yu. A. Kravtsov
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  2. Yu. I. Orlov
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© 1999 Springer-Verlag Berlin Heidelberg

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Kravtsov, Y.A., Orlov, Y.I. (1999). Typical Integrals of Catastrophe Theory. In: Caustics, Catastrophes and Wave Fields. Springer Series on Wave Phenomena, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59887-6_4

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  • DOI: https://doi.org/10.1007/978-3-642-59887-6_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64167-1

  • Online ISBN: 978-3-642-59887-6

  • eBook Packages: Springer Book Archive

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