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

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