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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