Phase-Space Differential Equations for Modes

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


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.


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 


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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

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