Multivalued solutions to the eikonal equation in stratified media
Authors:
S. Izumiya, G. T. Kossioris and G. N. Makrakis
Journal:
Quart. Appl. Math. 59 (2001), 365-390
MSC:
Primary 35Q60; Secondary 35B40, 35C20, 35J10, 58J47, 86A15
DOI:
https://doi.org/10.1090/qam/1828459
MathSciNet review:
MR1828459
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Abstract: In the present paper we study the geometric properties of the multivalued solutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating high frequency waves in stratified media. We consider the case of a fixed Hamiltonian imposed by the medium, and we present the geometric framework that describes the geometric solutions, using the notion of Legendrian immersions with an initial point source or an initial smooth front. Then, we study the singularities of the solutions in the case of a smooth or piecewise Hamiltonian in a boundaryless stratified medium. Finally, we study the singularities of the solutions in a domain with a boundary that describes the propagating field in a waveguide.
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G. Avila and J. Keller, The high-frequency asymptotic field of a point source in an inhomogeneous medium, Comm. Pure Appl. Math. XVI, 363–381 (1963)
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V. B. Andeev, A. V. Demin, Yu. A. Kravtsov, M. V. Tinin, and A. P. Yarygin, The interferential integral method (a review), Radiophys. Quantum Electron. 31(N11), 907–921 (1988)
J. M. Arnold, Oscillatory integral theory for uniform representation of wave functions, Rad. Sci. 17(5), 1181–1191 (1982)
J. M. Arnold, Spectral synthesis of uniform wavefunctions, Wave Motion 8, 135–150 (1986)
V. I. Arnol’d, A. N. Varchenko, and S. M. Guiseĭn-Zade, Singularities of Differentiate Maps, Vol. 1, Birkhäuser-Verlag, Basel, 1985
J. D. Benamou, Big ray tracing: Multivalued travel time field computation using viscosity solutions of the eikonal equation, J. Comput. Phys. 128, 463–474 (1996)
V. B. Babich, The short wave asymptotic form of the solution for the problem of a point source in an inhomogeneous medium, USSR J. Comput. Math. Phys. 5(5), 949–951 (1965)
V. B. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory. Asymptotic Methods, Springer-Verlag, Berlin-Heidelberg, 1991
J. D. Benamou, T. Katsaounis, and B. Perthame, High frequency Helmholtz equation, geometrical optics and particle methods, preprint (1998)
L. M. Brekhovskikh, Waves in Layered Media, Academic Press, New York, 1980
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V. M. Babich and N. Y. Kirpichnikova, The Boundary-Layer Method in Diffraction Problems, Springer-Verlag, Berlin-Heidelberg, 1979
I. A. Bogaevskiĭ, Perestroikas of fronts in evolutionary families, Proc. of the Steklov Inst. of Math. 209, 57–72 (1995)
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering, Springer-Verlag, New York, 1992
V. Cĕrvenỳ, I. A. Molotkov, and I. Ps̆enc̆ik, Ray Method in Seismology, Univerzita Karlova, Praha, 1977
J. J. Duistermaat, Oscillatory integrals, Lagrangian immersions and unfolding of singularities, Comm. Pure Appl. Math. XXVII, 207–281 (1974)
E. Fatemi, B. Engquist, and S. Osher, Numerical solution of the high frequency asymptotic expansion for the scalar wave equation, J. Comput. Phys. 120(1), 145–155 (1995)
Yu. L. Gazaryan, On the Ray Approximation near Nonsingular Caustics, Dynamic Theory of Seismic Wave Propagation 5, LGU Press, Leningrad, 1961, pp. 73–92
V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, RI, 1977
V. Guillemin and D. Schaeffer, Remarks on a paper of D. Ludwig, Bull. Amer. Math. Soc. 79(2), 382–385 (1973)
A. Hanyga, Canonical functions of asymptotic diffraction theory associated with symplectic singularities, Symplectic Singularities and Geometry of Gauge Fields, Banach Center Publications, Volume 36, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1997, pp. 57–71
A. Hanyga and M. Seredyńska, Diffraction of pulses in the vicinity of simple caustics and caustic cusps, Wave Motion 14, 101–121 (1991)
L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, New York, 1983
L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971)
S. Izumiya, Perestroikas of optical wave fronts and graphlike Legendrian unfoldings, J. Differential Geometry 38, 485–500 (1993)
K. Jänich, Caustics and catastrophes, Math. Ann. 209, 161–180 (1974)
D. S. Jones, High-frequency refraction and diffraction in general media, Philos. Trans. Roy. Soc. A 255, 341–387 (1963)
M. Kazarian, Caustics $D_{k}$ at points of interface between two media, Symplectic Geometry, London Math. Soc. Lecture Notes Series, vol. 192, 1993, pp. 115–125
T. Katsaounis, G. T. Kossioris, and G. N. Makrakis, Computation of high frequency fields near caustics, Tech. Rep. 98.7, IACM-FORTH (1998)
Yu. A. Kravtsov and Yu. I. Orlov, Caustics, Catastrophes and Wave Fields, Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1993
Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, Springer Series on Wave Phenomena, vol. 6, Springer-Verlag, Berlin, 1990
Yu. A. Kravtsov, Two new asymptotic methods in the theory of wave propagation in inhomogeneous media (review), Sov. Phys. Acoust. 14(1), 1–17 (1968)
V. V. Kucherenko, Quasiclassical asymptotics of a point-source function for the stationary Schrödinger equation, Theoret. Math. Phys. (English Translation) 1(3), 294–310 (1969)
D. Ludwig, Uniform asymptotic expansions at a caustic, Comm. Pure Appl. Math. XIX, 215–250 (1966)
V. V. Lychagin, Local classification of non-linear first order partial differential equations, Russian Math. Surveys 30, 105–175 (1975)
V. P. Maslov, Operational Methods, Mir Publishers, Moscow, 1976
V. P. Maslov, Theory of Perturbations and Asymptotic Methods, Dunod, Paris, 1972
V. P. Maslov and V. M. Fedoryuk, Semi-classical approximations in quantum mechanics, Contemp. Math. 5, D. Reidel, Dordrecht, 1981
A. Mishchenko, V. Shatalov, and B. Sternin, Lagrangian Manifolds and the Maslov Operator, Springer-Verlag, Berlin-Heidelberg, 1990
O. M. Myasnichenko, Singularities of wave fronts on the interface between two media, St. Petersburg Math. J. 5, 789–807 (1994)
O. Runborg, Multiscale and Multiphase Methods for Wave Propagation, Doctoral Dissertation, Dept. Num. Anal. Comp. Sci., Roy. Inst. Techn. Stockholm, 1998
I. G. Scherbak, Boundary fronts and caustics and their metamorphosis, Singularities (J.-P. Brasselet, ed.), London Math. Soc. Lecture Note Series 201, 1994, pp. 363–373
W. Symes, A slowness matching finite difference method for travel times beyond transmission caustics, Tech. Rep., Rice Univ., 1997
I. Tolstoy and C. S. Clay, Ocean Acoustics. Theory and Experiment in Underwater Sound, American Institute of Physics, New York, 1966
T. Tsukada, Reticular Lagrangian singularities, Asian J. Math. 1, 572–622 (1997)
T. Tsukada, Stability of optical caustics with r-corners, Hokkaido Math. J. 27, 633–650 (1998)
B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, 1989
B. R. Vainberg, Quasiclassical approximation in stationary scattering problems, Funct. Anal. Appl. 11, 247–257 (1977)
G. Wassermann, Stability of caustics, Math. Ann. 216, 43–50 (1975)
R. Weder, Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media, Springer-Verlag, New York, 1991
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