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Dynamics of wave propagation and curvature of discriminants

Victor P. Palamodov

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 6, page 1945-1981
  • ISSN: 0373-0956

Abstract

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For a Lagrange distribution of order zero we consider a quadratic integral which has logarithmic divergence at the singular locus of the distribution. The residue of the asymptotics is a Hermitian form evaluated in the space of positive distributions supported in the locus. An asymptotic analysis of the residue density is given in terms of the curvature form of the locus. We state a conservation law for the residue of the impulse-energy tensor of solutions of the wave equation which extends the classical conservation law in the geometrical optics.

How to cite

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Palamodov, Victor P.. "Dynamics of wave propagation and curvature of discriminants." Annales de l'institut Fourier 50.6 (2000): 1945-1981. <http://eudml.org/doc/75475>.

@article{Palamodov2000,
abstract = {For a Lagrange distribution of order zero we consider a quadratic integral which has logarithmic divergence at the singular locus of the distribution. The residue of the asymptotics is a Hermitian form evaluated in the space of positive distributions supported in the locus. An asymptotic analysis of the residue density is given in terms of the curvature form of the locus. We state a conservation law for the residue of the impulse-energy tensor of solutions of the wave equation which extends the classical conservation law in the geometrical optics.},
author = {Palamodov, Victor P.},
journal = {Annales de l'institut Fourier},
keywords = {Fourier integral; Lagrange manifold; contact bundle; symbol; discriminant; residue; curvature; conservation law},
language = {eng},
number = {6},
pages = {1945-1981},
publisher = {Association des Annales de l'Institut Fourier},
title = {Dynamics of wave propagation and curvature of discriminants},
url = {http://eudml.org/doc/75475},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Palamodov, Victor P.
TI - Dynamics of wave propagation and curvature of discriminants
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 6
SP - 1945
EP - 1981
AB - For a Lagrange distribution of order zero we consider a quadratic integral which has logarithmic divergence at the singular locus of the distribution. The residue of the asymptotics is a Hermitian form evaluated in the space of positive distributions supported in the locus. An asymptotic analysis of the residue density is given in terms of the curvature form of the locus. We state a conservation law for the residue of the impulse-energy tensor of solutions of the wave equation which extends the classical conservation law in the geometrical optics.
LA - eng
KW - Fourier integral; Lagrange manifold; contact bundle; symbol; discriminant; residue; curvature; conservation law
UR - http://eudml.org/doc/75475
ER -

References

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  1. [1] J.J. DUISTERMAAT, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math., 27 (1974), 207-278. Zbl0285.35010MR53 #9306
  2. [2] J.J. DUISTERMAAT, Fourier Integral Operators, Birkhäuser, 1996. Zbl0841.35137MR96m:58245
  3. [3] J.J. DUISTERMAAT, L. HÖRMANDER, Fourier integral operators, II, Acta Math., 128, 1-2 (1972), 183-269. Zbl0232.47055MR52 #9300
  4. [4] R. HARTSHORNE, Residues and Duality, Lectures Notes in Math., 20, Springer, 1966. Zbl0212.26101
  5. [5] L. HÖRMANDER, Fourier integral operators, I, Acta Math., 127, 1-2 (1971), 79-183. Zbl0212.46601MR52 #9299
  6. [6] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators, IV, Fourier Integral Operators, Springer, 1985. Zbl0601.35001
  7. [7] E. LOOIJENGA, The complement to the bifurcation variety of a simple singularity, Invent. Math., 23 (1974), 105-116. Zbl0278.32008
  8. [8] B. MALGRANGE, Ideals of differentiable functions, Oxford University Press, 1966. Zbl0177.17902
  9. [9] V.P. PALAMODOV, Distributions and Harmonic Analysis, Encyclopaedia of Mathematical Science, 72, Springer, 1993, 1-127. Zbl0826.46025MR1447631
  10. [10] L. SCHWARTZ, Théorie des distributions, Hermann, 1966. Zbl0149.09501
  11. [11] J.-C. TOUGERON, Idéaux de fonctions différentiables, Ann. Inst. Fourier, 18-1 (1968), 177-240. Zbl0188.45102MR39 #2171

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