Advertisement

Phase-Space Differential Equations for Modes

  • Leon Cohen
Part of the Operator Theory: Advances and Applications book series (OT, volume 205)

Abstract

We discuss how to transform linear partial differential equations into phase space equations. We give a number of examples and argue that phase space equations are more revealing than the original equations. Recently, phase space methods have been applied to the standard mode solution of differential equations and using this method new approximations have been derived that are better than the stationary phase approximation. The approximation methods apply to dispersion relations that exhibit propagation and attenuation. In this paper we derive the phase space differential equations that the approximations satisfy and also derive an exact phase space differential equation for a mode. By comparing the two we show that the approximations neglect higher-order derivatives in the phase space distribution.

Keywords

Phase space Wigner distributions Schrödinger free particle equation differential equation with drift modified diffusion equation linearized KdV equation 

Mathematics Subject Classification (2000)

Primary 35A22 35A27 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L. Cohen, Time-frequency distributions — A Review, Proc. IEEE, 77 (1989), 941–981.CrossRefGoogle Scholar
  2. [2]
    L. Cohen, Time-Frequency Analysis, Prentice-Hall, 1995.Google Scholar
  3. [3]
    L. Galleani and L. Cohen, The Wigner distribution for classical systems, Phys. Lett. A 302 (2002), 149–155.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    L. Galleani and L. Cohen, Direct time-frequency characterization of linear systems governed by differential Equations, Signal Processing Lett. 11 (2004), 721–724.CrossRefGoogle Scholar
  5. [5]
    L. Galleani and L. Cohen, Time-frequency Wigner distribution approach To differential equations, in Nonlinear Signal and Image Processing: Theory, Methods, and Applications, Editors: K. Barner and G. Arce, CRC Press, 2003.Google Scholar
  6. [6]
    P. Loughlin, Special issue on applications of time-frequency analysis, Proc. IEEE 84 (9), 1996.Google Scholar
  7. [7]
    P. Loughlin and L. Cohen, A Wigner approximation method for wave propagation, J. Acoust. Soc. Amer. 118 (3) (2005), 1268–1271.CrossRefGoogle Scholar
  8. [8]
    P. Loughlin and L. Cohen, Local properties of dispersive pulses, J. Modern Optics 49 (14/15) (2002), 2645–2655.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    P. H. Morse and K. U. Ingard, Theoretical Acoustics, McGraw-Hill, 1968.Google Scholar
  10. [10]
    J. E. Moyal, Quantum mechanics as a statistical Theory, Proc. Camb. Phil. Soc., 45 (1949), 99–124.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    I. Tolstoy and C. Clay, Ocean Acoustics: Theory and Experiment in Underwater Sound, AIP, 1987.Google Scholar
  12. [12]
    G. Whitham, Linear and Nonlinear Waves, J. Wiley and Sons, 1974.Google Scholar
  13. [13]
    E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932), 749–759.zbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Leon Cohen
    • 1
  1. 1.Department of Physics and AstronomyCity University of New York Hunter CollegeNew YorkUSA

Personalised recommendations