362 Index

ray cone, 77, 78

ray tracing algorithms, 166

rays, 144–145

and conormal waves, xviii

spread of, 25

transport along, 2, 23, 161–177,

285–289

tube of, 26, 34, 100, 147, 148, 288

rectilinear propagation, xvi, 20–27

reflection

coeﬃcient of, 32, 35, 36

law of, xvi, 28

operator, 30

total, 42

refraction, Snell’s law, xvi, 36–42

resonance, xviii

collinear, 341

examples of, 291–302, 317–332

introduction to, 291–294

quadratic, definition, 341

quasilinear, 302–314, 327–332

relation, 292

relations for Euler equations, 341–342

semilinear, 291–302, 314–315,

321–327

R¨ omer, xvi

Schauder’s lemma, 217–222

Schr¨ odinger’s equation, 149–151

self phase modulation, 287, 288

semiclassical limit of quantum

mechanics, 278

semisimple eigenvalue, 117

short wavelength asymptotic analysis,

xvi

singularities

(microlocal) of piecewise smooth

functions, 140

and the method of characteristics,

10–11

for progressing waves, 12–16

propagation for piecewise smooth

waves, 10–11, 194

propagation global in time, 192–195

propagation local in time, 188–191

propagation of, xiv, xviii

propagation of and stabilization,

195–205

slowly varying envelope approximation,

287

small divisor, 310–313

hypothesis, 312, 318

for Euler equations, 339

smooth characteristic variety

hypothesis, 163–178, 286–289

smooth points, of the characteristic

variety, 76–78

Snell’s law, see refraction

Sobolev embedding, 216, 220

space like, 146

spectral projection, 117

spectrum

of a periodic function, 309

of F (V ), 315

of principal profile, 309

stability

Hadamard’s notion of, 43

theorem, quasilinear, 314

theorem, semilinear, 283

stationary phase, 210

stationary phase inequality, 120–122

stationary point, nondegenerate, 120

stratification theorem, 76, 94, 99

Strichartz inequalities, xii, 111–117,

251–257

three wave interaction

infinite system, 337

ode, 298–302, 330–332, 336

pde, 294–298, 322–327

resonance of order three, 341

time of nonlinear interaction

quasilinear, 303

semilinear, 262

time-like, 146

cone, 64–71

transport equation, 144, 146

wave

acoustic, xiv, 341

conormal, xviii, 16, 194

plane, 14, 33–34, 71–79, 150

progressing, 12–16

shock, xv, 12

short pulse, xvi

spherical, 32, 171–173

vorticity, 341

wave packet, xiv, 1, 22–28, 34–36, 142

wave train, 1, 291

wave number, good and bad, 93–96

wavefront set, 136–140, 186–195,

201–204

Wiener algebra, 96