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Asymptotic solution of the Cauchy problem for equations with complex characteristics

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Abstract

Differential equations with a small parameter in the derivative are considered. A method is developed for constructing formal asymptotic solutions for the case of complex characteristics. For this a new class of manifolds is introduced which is a natural generalization of real Lagrangian manifolds to the complex case. The theory of the canonical Maslov operator is constructed in this class of manifolds. Asymptotic solutions are expressed in terms of the canonical Maslov operator.

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Literature cited

  1. V. I. Arnol'd, “On the characteristic class contained in the quantization condition,” Funkts. Anal. Prilozhen.,1, No. 1, 1–14 (1967).

    Google Scholar 

  2. V. I. Arnol'd, Lectures on Classical Mechanics [in Russian], Moscow State Univ. (1968).

  3. V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Problems of the Diffraction of Short Waves. The Method of Comparison Problems [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  4. V. S. Buslaev, “The generating integral and the canonical Maslov operator in the WKB method,” Funkts. Anal. Prilozhen.,3, No. 3, 17–31 (1969).

    Google Scholar 

  5. M. I. Vishik and L. A. Lyusternik, “Regular degeneration and the boundary layer for linear differential equations with a small parameter,” Usp. Mat. Nauk,12, No. 5, 3–122 (1957).

    Google Scholar 

  6. S. G. Gindikin and M. V. Fedoryuk, “Asymptotics of the fundamental solution for a differential equation with constant coefficients which is parabolic in the sense of Petrovskii,” Mat. Sb.,91, No. 4, 500–522 (1973).

    Google Scholar 

  7. V. G. Danilov, “On the boundedness of pseudodifferential operators in Sobolev spaces,” Mosk. Inst. Elektron. Mashinostr., Moscow (1974) (manuscript submitted to VINITI May 22, 1945, No. 1383-74 Dep).

    Google Scholar 

  8. N. Dunford and J. Schwartz, Linear Operators. General Theory [Russian translation], Part 1, IL, Moscow (1962).

    Google Scholar 

  9. M. A. Evgrafov and M. M. Postnikov, “Asymptotics of Green functions for parabolic and elliptic equations with constant coefficients,” Mat. Sb.,82, No. 1, 3–29 (1970).

    Google Scholar 

  10. V. V. Kucherenko, “Quasiclassical asymptotics of the function of a point source for the stationary Schrödinger equation,” Teor. Mat. Fiz.,1, No. 3, 384–406 (1969).

    Google Scholar 

  11. V. V. Kucherenko, “The Hamilton-Jacobi equation in the complex, nonanalytic situation,” Dokl. Akad. Nauk SSSR,213, No. 5, 1021–1024 (1973).

    Google Scholar 

  12. V. V. Kucherenko, “The canonical Maslov operator on the germ of a complex almost analytic manifold,” Dokl. Akad. Nauk SSSR,213, No. 6, 1251–1254 (1973).

    Google Scholar 

  13. V. V. Kucherenko, “The commutation formula of an h−1-pseudodifferential operator with rapidly oscillating exponential in the case of complex phase,” Mat. Sb.,94, No. 1, 89–113 (1974).

    Google Scholar 

  14. V. V. Kucherenko, “Asymptotics of the solution of the system A(x, −ih(δ/δx))u=0 as h-0 in the case of characteristics of variable multiplicity,” Izv. Akad. Nauk SSSR, Ser. Mat.,38, No. 3, 625–662 (1974).

    Google Scholar 

  15. V. V. Kucherenko, “Asymptotic solutions of equations with complex characteristics,” Mat. Sb.,95, No. 2, 163–213 (1974).

    Google Scholar 

  16. J. Leray, L. Gårding, and T. Kotake, The Cauchy Problem. Uniformization and Asymptotic Expansion of the Solution of a Linear Cauchy Problem with Holomorphic Data. Analogy with the Theory of Asymptotic and Approximate Waves [Russian translation], Mir, Moscow (1967).

    Google Scholar 

  17. V. P. Maslov, Perturbation Theory and Asymptotic Methods [in Russian], Moscow State Univ. (1965).

  18. V. P. Maslov, “The canonical operator on a Lagrangian manifold with complex germ and the regularizer for pseudodifferential operators and difference schemes,” Dokl. Akad. Nauk SSSR,195, No. 3, 551–554 (1970).

    Google Scholar 

  19. V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  20. V. P. Maslov and V. G. Danilov, “Quasiinvertibility of functions of ordered operators in the theory of pseudodifferential equations,” in: Sovrem. Probl. Mat., Vol. 6 (Itogi Nauki i Tekhn. VINITI AN SSSR), Moscow (1976), pp. 5–132.

    Google Scholar 

  21. V. P. Maslov and Yu. Yu. Sternin, “The canonical operator (the complex case),” in: Sovrem. Probl. Mat., Vol. 1 (Itogi Nauki i Tekhn. VINITI AN SSSR), Moscow (1973), pp. 169–195.

    Google Scholar 

  22. V. P. Maslov and M. V. Fedoryuk, “The canonical operator (the real case),” in: Sovrem. Probl. Mat., Vol. 1 (Itogi Nauki i Tekhn. VINITI AN SSSR), Moscow (1973), pp. 85–167.

    Google Scholar 

  23. J. Miinor, Morse Theory [Russian translation], Mir, Moscow (1965).

    Google Scholar 

  24. A. S. Mishchenko, B. Yu. Sternin, and V. E. Shatalov, The Method of the Canonical Maslov Operator (Complex Theory), MIEM (1974).

  25. A. G. Prudkovskii, “The method of stationary phase in application to integrals depending on a parameter,” Zh. Vychisl. Mat. Mat. Fiz.,13, No. 2, 275–293 (1973).

    Google Scholar 

  26. A. G. Prudkovskii, “The method of stationary phase in application to integrals depending on a parameter (nonanalytic case),” Zh. Vychisl. Mat. Mat Fiz.,4, No. 2, 299–311 (1974).

    Google Scholar 

  27. B. Yu. Sternin and V. E. Shatalov, “The smooth theory of the canonical Maslov operator on a complex Lagrangian germ,” Usp. Mat. Nauk,29, No. 3, 229–230 (1974).

    Google Scholar 

  28. M. V. Fedoryuk, “The method of stationary phase and pseudodifferential operators,” Usp. Mat. Fiz.,26, No. 1, 67–112 (1971).

    Google Scholar 

  29. L. Hörmander, “Fourier integral operators. I,” Matematika, Period. Sb. Perev. In. Statei,16, No. 1, 17–61 (1972).

    Google Scholar 

  30. L. Hörmander, “Fourier integral operators. II,” Matematika, Period. Sb. Perev. In. Statei,16, No. 2, 67–136 (1972).

    Google Scholar 

  31. V. Maslov, “The characteristics of pseudodifferential operators and difference schemes,” Actes Congr. Int. Mathematiciens, 1970, Vol. 2, Paris (1971), pp. 755–769.

    Google Scholar 

  32. F. Treves, “Hypoelliptic partial differential equations of principal type. Sufficiency and necessary conditions,” Commun. Pure Appl. Math.,24, No. 5, 631–670 (1971).

    Google Scholar 

  33. J. Sjöstrand and A. Melin, “Fourier integral operators with complex-valued phase functions,” Proceedings of the International Conference in Nice on Fourier Integral Operators (1974).

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Authors
  1. V. V. Kucherenko
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Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 41–136, 1977.

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Kucherenko, V.V. Asymptotic solution of the Cauchy problem for equations with complex characteristics. J Math Sci 13, 24–81 (1980). https://doi.org/10.1007/BF01084109

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