360 Index
characteristic polynomial, 65
Euler’s equations, 334
Maxwell’s equations, 67, 72
characteristic variety, 65, 163
curved and flat, 95
commutator, 8, 233–235, 281–282
E, 271
computer approximations, xv, xvii, 166
conservation of charge, 44
constant rank hypothesis, 154–155
continuity equation, 44
controlability, xviii, 195
corrector, 26, 126, 310
curved sheets, of characteristic variety,
95
D’Alembert’s formula, 3, 13, 253
diffractive geometric optics, xii, 25
dimensional analysis, 221
dispersion, 92, 111
dispersive behavior, 91–117, 149
relation, 14, 23, 34, 36, 78, 92
Schr¨ odinger equation, 150
dispersive geometric optics, xii, xvii,
261
domains of influence and determinacy,
58–59, 70–71
Duhamel’s formula, 57
eikonal equation
and constant rank hypothesis, 178
and Lax parametrix, 177
and three wave interaction, 292–294
for nonlinear geometric optics, 266
Hamilton-Jacobi theory for, 206–214
Schr¨ odinger equation, 150–151
simple examples, 152–155
elliptic operator, xiv, 79
elliptic regularity theorem, 132–140
elliptic case, 6
microlocal, 136–140
emission, 79–83
energy
conservation of, 13, 14, 45, 147,
169–177, 246
method, 29–30, 36–38
energy, conservation of, 78
Euler equations
compressible inviscid, 310
dense oscillations for, 333–350
Fermat’s principle of least time, xvi
finite speed, xiv, 10, 29, 38, 58–83
for semilinear equations, 224
speed of sound, 341
flat parts, of characteristic variety, 95
Fourier integral operator, xii, 177,
180–188
Fourier transform, definition, 45
frequency conversion, 302
fundamental solution, 13–15, 20
geometric optics, xi, xv, xvi
cautionary example, 27
elliptic, 123–132
from solution by Fourier transform,
20–27
linear hyperbolic, 141–177
nonlinear multiphase, 291–350
nonlinear one phase, 259, 277–289,
310
physical, 1, 16
second order scalar, 143–149
Gronwall’s Lemma, 50
group velocity, 34, 36
and decay for maximally dispersive
systems, 111
and Hamilton-Jacobi theory, 210
and nonstationary phase, 16–20
and smooth variety hypothesis, 163
conormal to characteristic variety,
71–78
for anisotropic wave equation, 27
for curved sheets of characteristic
variety, 100
for D’Alembert’s equation, 23
for scalar second order equations,
145–149
for Schr¨ odinger’s equation, 149
Guoy shift, 173
Haar’s inequality, 7–12
Hamilton–Jacobi theory
for hyperbolic problems, 152–155
Schr¨ odinger equation, 150–151
Hamilton-Jacobi theory, 206–214
harmonics, generation of, xviii, 260–263
homogeneous Sobolev norm, 112
hyperbolic
constant coefficient, 84–89
constant multiplicity, 89, 178
strictly, xiv, 6, 89
second order, 146
symmetric, 44–58, 88, 151–195
definition constant coefficients, 45
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