Gaussian beams summation for the wave equation in a convex domain

Communications in Mathematical Sciences

Volumn 7, Issue 4, 2009, Pages 973-1008

  Bougacha, Salma ^a   Akian, Jean Luc ^b   Alexandre, Radjesvarane ^c  

^a UNIVERSITÉ D EVRY VAL D ESSONNE   (France)

^b ONERA   (France)

^c IRENav   (France)

Author keywords

Gaussian beams summation; High frequency solutions; Reflection at the boundary; Wave equation

Indexed keywords

EID: 77953933814     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/cms.2009.v7.n4.a9     Document Type: Article

References (45)

1. R. Alexandre, S. Bougacha and J.L. Akian, Gaussian beams summation for computing high frequency waves, preprint.
2. V.M. Babich, Eigenfunctions concentrated in a neighborhood of a closed geodesic, V.M. Babich (eds.), Math. Problems in Wave Propagation Theory, Sem. Math., V.A. Steklov Math. Inst., 9, 1968. Translated by Consultants Bureau, New York, 1970.
3. H. Beirão da Veiga, Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations, Commun. Pure Appl. Math., 46, 221-259, 1993.
4. J. Butler, Global h-Fourier integral operators with complex-valued phase functions, Bull. Lond. Math. Soc., 34, 479-489, 2002.
5. V. Červený, M.M. Popov and I. Pšenčík, Computation of wave elds in inhomogeneous media-Gaussian beam approach, Geophys. J. R. Astr. Soc., 70, 109-128, 1982.
6. J. Chazarain, Paramétrix du probléme mixte pour l'équation des ondes à l'intérieur d'un do-maine convexe pour les bicaract́eristiques, Ast́erisque, 34-35, 165-181, 1976.
7. J. Chazarain and A. Piriou. Introduction to the Theory of Linear Partial Differential Equations, North-Holland Publishing Co., Amsterdam, 1982.
8. J.-M. Delort, F.B.I.Ttransformation. Second Microlocalization and Semilinear Caustics, Lec-ture Notes in Mathematics, Springer-Verlag, Berlin, 1522, 1992.
9. J.J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Commun. Pure Appl. Math., 27, 207-281, 1974.
10. L.B. Felsen, Real spectra, complex spectra, compact spectra, J. Opt. Soc. Amer. A, 3, 486-496, 1986.
11. L.E. Fraenkel, Formulae for high derivatives of composite functions, Math. Proc. Cambridge Philos. Soc., 83, 159-165, 1978.
12. P. Ǵerard, P.A. Markowich, N.J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Commun. Pure Appl. Math., 50, 323-379, 1997.
13. J.K. Hale, Ordinary Differential Equations, R.E. Krieger Pub. Co., Huntington, New York, 1980.
14. N.R. Hill, Prestack Gaussian-beam depth migration, Geophysics, 66, 1240-1250, 2001.
15. L. Hörmander, Fourier integral operators I, Acta Math., 127, 79-83, 1971.
16. L. Hörmander, On the existence and the regularity of solutions of linear pseudo-differential equations, L'Enseignement Mathématique, 17, 99-163, 1971.
17. L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer-Verlag, Berlin, 1985.
18. S. Jin, H. Wu and X. Yang, Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Commun. Math. Sci., 6, 995-1020, 2008.
19. A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC, Boca Raton, 123, 2001.
20. A.P. Katchalov and M.M. Popov, The application of the Gaussian beam summation method to the computation of high-frequency wave elds, Dokl. Akad. Nauk, 258, 1097-1100, 1981.
21. J.B. Keller, Corrected Bohr-Sommerfeld quantum conditions for non separable systems, Ann. Physics, 4, 180-188, 1958.
22. L. Klimeš, Expansion of a high-frequency time-harmonic waveeld given on an initial surface into Gaussian beams, Geophys. J. R. astr. Soc., 79, 105-118, 1984.
23. A. Laptev and Y. Safarov, Global solution of the wave equation, in Journées Équations aux dérivées partielles, Saint-Jean-De-Monts, France, 1-9, 1990.
24. A. Laptev and I.M. Sigal, Global Fourier integral operators and semiclassical asymptotics, Rev. Math. Phys., 12 749-766, 2000.
25. I. Lasiecka, J.L. Lions and R. Triggiani, Non-homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65, 149-192, 1986.
26. S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, J. Comput. Phys., 228 2951-2977, 2009.
27. J.L. Lions and E. Magenes, Problémes aux limites non homogénes et applications, Dunod, Paris, 1, 1968.
28. H. Liu and J. Ralston, Recovery of high frequency wave elds for the acoustic wave equation, preprint, 2009.
29. H. Liu and J. Ralston, Recovery of high frequency wave elds from phase space based measure-ments, preprint, 2009.
30. F. Macía and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach, Asymptot. Anal., 32, 1-26, 2002.
31. A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Springer-Verlag, New York, 2002.
32. V.P. Maslov and M.V. Fedoruk, Semiclassical Approximation in Quantum Mechanics, Reidel, Dordrecht, 1981.
33. R.B. Melrose, Microlocal parametrices for diffractive boundary value problems, Duke Math. J., 42, 605-635, 1975.
34. M. Motamed and O. Runborg, Taylor expansion errors in Gaussian beam summation, preprint, 2008.
35. T. Paul and A. Uribe, On the pointwise behavior of semi-classical measures, Commun. Math. Phys., 175, 229-258, 1996.
36. J. Ralston, Gaussian beams and the propagation of singularities, Studies in PDEs, MAA stud. Math., 23, 206-248, 1982.
37. J. Ralston, Gliding beams and whispering gallery quasi-modes, Michigan State, March, 2008.
38. J. Rauch and K. Markus, Lectures on geometric optics, L. Caffarelli (eds.), in Hyperbolic Equations and Frequency Interactions, IAS/Park City Math. Ser., 5, 383, 1999.
39. S.L. Robinson, Semiclassical mechanics for time-dependent Wigner functions, J. Math. Phys., 34, 2185-2205, 1993.
40. 2-boundedness theorems for semiclassical Fourier integral operators with complex phase, preprint, 2007.
41. 1,1 coeffcients, Inverse Probl. Imaging, 1, 537-556, 2007.
42. J. Sjöstrand, Singularités analytiques microlocales, Astérisque, Soc. Math. de France, 95, 1982.
43. T. Swart and V. Rousse, A mathematical justication for the Herman-Kluk propagator, Commun. Math. Phys., 286, 725-750, 2009.
44. N.M. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6, 449-475, 2008.
45. N.M. Tanushev, J. Qian and J.V. Ralston, Mountain waves and Gaussian beams, Multiscale Model. Simul., 6, 688-709, 2007.