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