Exact equations for smoothed Wigner transforms and homogenization of wave propagation

Applied and Computational Harmonic Analysis

Volumn 24, Issue 3, 2008, Pages 378-392

  Athanassoulis, Agissilaos G ^a  

^a PRINCETON UNIVERSITY   (United States)

Author keywords

Cohen's class; High frequency computational techniques; Phase space methods for wave propagation; Wigner transform

Indexed keywords

EID: 41349085351     PISSN: 10635203     EISSN: 1096603X     Source Type: Journal    
DOI: 10.1016/j.acha.2007.06.006     Document Type: Letter

References (43)

1. Arnold A., Carrillo J.A., Gamba I., and Shu C. Low and high field scaling limits for the Vlasov- and Wigner-Poisson-Fokker-Planck systems. Transport Theory Statist. Phys. 30 (2001) 121-153
2. A.G. Athanassoulis, Smoothed Wigner transforms and homogenization of wave propagation, Ph.D. thesis; temporarily available at http://www.math.princeton.edu/~aathanas/mns81/thesis.zip
3. A.G. Athanassoulis, Smoothed Wigner transforms in the numerical simulation of semiclassical (high-frequency) wave propagation, Dyn. Contin. Discrete Impuls. Syst. Ser. A (Special Issue), in press
4. Belov V.V., Dobrokhotov S.Y., Sinitsy S.O., and Ya T. Maslov's canonical operator for nonrelativistic equations of quantum mechanics in curved nanotubes. Dokl. Ross. Akad. Nauk 68 (2003) 460-465
5. Benamou J.D., Castella F., Katsaounis T., and Perthame B. High frequency limit of the Helmholtz equation. Rev. Mat. Iberoamericana 18 (2002) 187-209
6. Cohen L., and Galleani L. Nonlinear transformation of differential equations into phase space. EURASIP J. Appl. Signal Process. 12 (2004) 1770-1777
7. Cohen L. Time-Frequency Analysis (1995), Prentice Hall PTR
8. Cohen L. Generalized phase space distributions. J. Math. Phys. 7 (1966) 781-786
9. Engquist B., and Runborg O. Computational high frequency wave propagation. Acta Numer. 12 (2003) 181-266
10. Fannjiang A.C. Nonlinear Schrödinger equation with a white-noise potential: Phase-space approach to spread and singularity. Phys. D 212 (2005) 195-204
11. Fannjiang A.C. White-noise and geometrical optics limits of Wigner-Moyal equation for beam waves in turbulent media II: Two-frequency formulation. J. Stat. Phys. 120 (2005) 543-586
12. Fannjiang A.C., Jin S., and Papanicolaou G. High frequency behavior of the focusing nonlinear Schrödinger equation with random inhomogeneities. SIAM J. Appl. Math. 63 (2003) 1328-1358 (eetron)
13. Fermanian Kammerer C., and Gerard P. A Landau-Zener formula for non-degenerated involutive codimension 3 crossings. Ann. Henri Poincare 4 3 (2003) 513-552
14. Filippas S., and Makrakis G.N. Semiclassical Wigner function and geometrical optics. Multiscale Model. Simul. 1 (2003) 674-710 (eetron)
15. Flandrin P. Time-Frequency/Time-Scale Analysis (1999), Academic Press, San Diego, CA
16. Flandrin P., and Martin W. The Wigner-Ville spectrum of nonstationary random signals. In: Mecklenbrauker W., and Hlawatsch F. (Eds). The Wigner Distribution (1997), Elsevier, Amsterdam 211-267
17. Folland G.B. Harmonic Analysis in Phase Space (1989), Princeton University Press, Princeton, NJ
18. Gerard P. A microlocal version of concentration-compactness. Partial Differential Equations and Mathematical Physics. Progr. Nonlinear Differential Equations Appl. vol. 21 (1996), Birkhäuser, Boston 143-157
19. Gerard P. Microlocal defect measures. Comm. Partial Differential Equations 16 (1991) 1761-1794
20. Gerard P., Markowich P.A., Mauser N.J., and Poupaud F. Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50 (1997) 323-379
21. Grossmann A., Loupias G., and Stein E.M. An algebra of pseudodifferential operators and quantum mechanics in phase space. Ann. Inst. Fourier (Grenoble) 18 (1968) 343-368
22. Hall B., Lisak M., Anderson D., Fedele R., and Semenov V.E. Statistical theory for incoherent light propagation in nonlinear media. Phys. Rev. E 65 (2002)
23. Hlawatsch F., and Flandrin P. The interference structure of the Wigner distribution and related time-frequency signal representations. In: Mecklenbrauker W., and Hlawatsch F. (Eds). The Wigner Distribution (1997), Elsevier, Amsterdam 59-133
24. Jaffard S., Meyer Y., and Ryan R.D. Wavelets (2001), SIAM, Philadelphia, PA
25. Janssen A.J.E.M. Positivity and spread of bilinear time-frequency distributions. In: Mecklenbrauker W., and Hlawatsch F. (Eds). The Wigner Distribution (1997), Elsevier, Amsterdam 1-58
26. Jin S., and Li X. Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs. Wigner. Phys. D 182 (2003) 46-85
27. Jin S., Liu H., Osher S., and Tsai R. Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems. J. Comput. Phys. 210 (2005) 497-518
28. Jin S., Liu H., Osher S., and Tsai R. Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation. J. Comput. Phys. 205 (2005) 222-241
29. Lasater M.S., Kelley C.T., Salinger A.G., Woolard D.L., and Zhao P. Parallel parameter study of the Wigner-Poisson equations for RTDs. Comput. Math. Appl. 51 (2006) 1677-1688
30. Lions P.L., and Paul T. Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9 (1993) 553-618
31. Markowich P.A., Mauser N.J., and Poupaud F. A Wigner-function approach to (semi)classical limits: Electrons in a periodic potential. J. Math. Phys. 35 (1994) 1066-1094
32. Mauser N.J. (Semi)classical limits of Schrödinger-Poisson systems via Wigner transforms. Journees Equations aux Derivees Partielles (2002), Univ. Nantes Ep. No. 12
33. Moyal J.E. Quantum mechanics as a statistical theory. Proc. Cambridge Philos. Soc. 45 (1949) 99-124
34. Muratov B. On the well-posedness of equations for smoothed phase space distribution functions and irreversibility in classical statistical mechanics. J. Phys. A 34 (2001) 4641-4651
35. Ozdemir K., and Arikan O. Fast computation of the ambiguity function and the Wigner distribution on arbitrary line segments. IEEE Trans. Signal Process. 49 (2001) 381-393
36. Pool J.C.T. Mathematical aspects of the Weyl correspondence. J. Math. Phys. 7 (1966) 66-76
37. Ryzhik L., Papanicolaou G., and Keller J.B. Transport equations for elastic and other waves in random media. Wave Motion 24 (1996) 327-370
38. Sparber, Markowich P.A., and Mauser N.J. Wigner functions versus WKB-methods in multivalued geometrical optics. Asymptot. Anal. 33 (2003) 153-187
39. Tatarskii V.I. The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26 (1983) 311-327
40. Whitham G.B. Linear and Nonlinear Waves (1999), John Wiley, New York
41. Wigner E. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40 (1932) 749
42. Ying L., and Candes E. The phase flow method. J. Comput. Phys. 220 (2006) 184-215
43. Zhang P., Zheng Y., and Mauser N.J. The limit from the Schrödinger-Poisson to the Vlasov-Poisson equations with general data in one dimension. Comm. Pure Appl. Math. 55 (2002) 582-632