2 1. Simple Examples of Propagation by such solutions have their singular behavior spread over very small lengths, not exactly zero. The path of a localized structure in space-time is curvelike, and such curves are often called rays. When phenomena are described by partial dif- ferential equations, linking the above ideas with the equation means finding solutions whose salient features are localized and in simple cases are de- scribed by transport equations along rays. For wave packets such results appear in an asymptotic analysis as 0. In this chapter some introductory examples are presented that illustrate propagation of singularities, propagation of energy, group velocity, and short wavelength asymptotics. That energy and singularities may behave very differently is a consequence of the dichotomy that, up to an error as small as one likes in energy, the data can be replaced by data with compactly supported Fourier transform. In contrast, up to an error as smooth as one likes, the data can be replaced by data with Fourier transform vanishing on |ξ| R with R as large as one likes. Propagation of singularities is about short wavelengths while propagation of energy is about wavelengths bounded away from zero. When most of the energy is carried in short wavelengths, for example the wave packets above, the two tend to propagate in the same way. 1.1. The method of characteristics When the space dimension is equal to one, the method of characteristics reduces many questions concerning solutions of hyperbolic partial differential equations to the integration of ordinary differential equations. The central idea is the following. When c(t, x) is a smooth real valued function, introduce the ordinary differential equation (1.1.1) dx dt = c(t, x) . Solutions x(t) satisfy dx(t) dt = c(t, x(t)). For a smooth function u, d dt u(t, x(t)) = ( ∂tu + c ∂xu ) (t,x(t)) . Therefore, solutions of the homogeneous linear equation ∂tu + c(t, x) ∂xu = 0 are exactly the functions u that are constant on the integral curves (t, x(t)). These curves are called characteristic curves or simply characteristics.
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