Diffractive behavior of the wave equation in periodic media: Weak convergence analysis

Annali di Matematica Pura ed Applicata

Volumn 188, Issue 4, 2009, Pages 561-589

  Allaire, Grégoire ^a   Palombaro, Mariapia ^b   Rauch, Jeffrey ^c  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

^c UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Bloch waves; Diffractive geometric optics; Homogenization

Indexed keywords

EID: 70349652355     PISSN: 03733114     EISSN: 16181891     Source Type: Journal    
DOI: 10.1007/s10231-008-0089-y     Document Type: Article

References (31)

1. Allaire G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482-1518 (1992).
2. Allaire G.: Dispersive limits in the homogenization of the wave equation. Annales de la Faculté des Sciences de Toulouse XII, 415-431 (2003).
3. Allaire, G., Amar, M.: Boundary layer tails in periodic homogenization, ESAIM COCV 4, pp. 209-243 (1999).
4. Allaire G., Capdeboscq Y., Piatnistki A., Siess V., Vanninathan M.: Homogenization of periodic systems with large potentials. Archive Rat. Mech. Anal. 174, 179-220 (2004).
5. Allaire, G., Palombaro, M., Rauch, J.: Diffractive geometric optics for Bloch wave packets, preprint.
6. Allaire G., Piatnistki A.: Homogenization of the Schrödinger equation and effective mass theorems. Comm. Math Phys. 258, 1-22 (2005).
7. Bamberger A., Engquist B., Halpern L., Joly P.: Parabolic wave equation approximations in heterogenous media. SIAM J. Appl. Math. 48(1), 99-128 (1988).
8. Benisty, H., Weisbuch, C.: Photonic crystals. In: Wolf, E. (ed.) Progress in optics, vol. 49, pp. 177-313. Elsevier, Amsterdam (2006).
9. Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures. North-Holland, Amsterdam (1978).
10. Brahim-Otsmane S., Francfort G., Murat F.: Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 71(9), 197-231 (1992).
11. Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam (1973).
12. Conca C., Planchard J., Vanninathan M.: Fluids and periodic structures, RMA, vol. 38. Wiley & Masson, Paris (1995).
13. Conca, C., Orive, R., Vanninathan, M.: On Burnett coefficients in strongly periodic media (preprint).
14. Donnat, P., Joly, J.-L., Metivier, G., Rauch, J.: Diffractive nonlinear geometric optics, Séminaire sur les Equations aux Dérivées Partielles, 1995-1996, Exp. No. XVII, p. 25. Ecole Polytechnique, Palaiseau (1996).
15. Donnat, P., Rauch, J.: Modeling the dispersion of light, Singularities and oscillations (Minneapolis, MN, 1994/1995), IMA Vol. Math. Appl., vol. 91, pp. 17-35. Springer, New York (1997).
16. Donnat P., Rauch J.: Dispersive nonlinear geometric optics. J. Math. Phys. 38(3), 1484-1523 (1997).
17. Francfort G., Murat F.: Oscillations and energy densities in the wave equation. Comm. Partial Differ. Equ. 17, 1785-1865 (1992).
18. Gérard P.: Microlocal defect measures. Comm. Partial Differ. Equ. 16, 1761-1794 (1991).
19. Gérard P., Markowich P., Mauser N., Poupaud F.: Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50(4), 323-379 (1997).
20. Kato T.: Perturbation theory for linear operators. Springer, Berlin (1966).
21. Kuchment, P.: Floquet theory for partial differential equations, Operator Theory: Advances and Applications, vol. 60. Birkhäuser Verlag, Basel (1993).
22. Kuchment, P.: The mathematics of photonic crystals, Mathematical modeling in optical science, Frontiers Appl. Math., vol. 22, pp. 207-272. SIAM, Philadelphia (2001).
23. Leontovich M., Fock V.: Solution of the problem of propagation of electromagnetic waves along the earth's surface by the method of parabolic equation. Acad. Sci. USSR. J. Phys. 10, 13-24 (1946).
24. de Martin Sterke C., Sipe J.: Envelope-function approach for the electrodynamics of nonlinear periodic structures. Phys. Rev. A 38, 5149-5165 (1998).
25. Marusic-Paloka E., Piatnitski A.: Homogenization of nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection. J. Lond. Math. Soc. 72(2), 391-409 (2005).
26. Nguetseng G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608-623 (1989).
27. Reed M., Simon B.: Methods of Modern Mathematical Physics. Academic Press, New York (1978).
28. Santosa F., Symes W.: A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51, 984-1005 (1991).
29. Sipe J., Winful H.: Nonlinear Schrödinger solitons in a periodic structure. Opt. Lett. 13, 132-133 (1988).
30. Tappert, F.: The parabolic approximation method, in Wave propagation and underwater acoustics (Workshop, Mystic, Conn., 1974). Lecture Notes in Physics, vol. 70, pp. 224-287. Springer, Berlin (1977).
31. Tartar L.: H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. R. Soc. Edinburgh 115A, 193-230 (1990).