Phase-Space Differential Equations for Modes

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.