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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