AN ELEMENTARY TREATISE ON GEOMETRICAL OPTICS
CONTENTS CHAPTER 1 SIMPLE EXAMPLES OF PROPAGATION §1 1 THE METHOD OF CHARACTERISTICS §1 2 EXAMPLES OF PROPAGATION OF SINGULARITIES USING PROGRESSING WAVES §1 3 GROUP VELOCITY AND THE METHOD OF NONSTATIONARY PHASE §1 4 FOURIER SYNTHESIS AND RECTILINEAR PROPAGATION §1 5 A CAUTIONARY EXAMPLE IN GEOMETRIC OPTICS §1 6 THE LAW OF RE?ECTION 1 6 1 THE METHOD OF IMAGES 1 6 2 THE PLANE WAVE DERIVATION 1 6 3 RE?ECTED HIGH FREQUENCY WAVE PACKETS §1 7 SNELL’S LAW OF REFRACTION CHAPTER 2 THE LINEAR CAUCHY PROBLEM §2 1 ENERGY ESTIMATES FOR SYMMETRIC HYPERBOLIC SYSTEMS §2 2 EXISTENCE THEOREMS FOR SYMMETRIC HYPERBOLIC SYSTEMS §2 3 FINITE SPEED OF PROPAGATION 2 3 1 THE METHOD OF CHARACTERISTICS 2 3 2 SPEED ESTIMATES UNIFORM IN SPACE 2 3 3 TIME-LIKE AND PROPAGATION CONES §2 4 PLANE WAVES, GROUP VELOCITY, AND PHASE VELOCITIES §2 5 PRECISE SPEED ESTIMATE §2 6 LOCAL CAUCHY PROBLEMS APPENDIX 2 I CONSTANT COE?CIENT HYPERBOLIC SYSTEMS APPENDIX 2 II FUNCTIONAL ANALYTIC PROOF OF EXISTENCE CHAPTER 3 DISPERSIVE BEHAVIOR §3 1 ORIENTATION §3 2 SPECTRAL DECOMPOSITION OF SOLUTIONS §3 3 LARGE TIME ASYMPTOTICS §3 4 MAXIMALLY DISPERSIVE SYSTEMS 3 4 1 THE L1 ?L?DECAY ESTIMATE 3 4 2 FIXED TIME DISPERSIVE SOBOLEV ESTIMATES 3 4 3 STRICHARTZ ESTIMATES APPENDIX 3 I PERTURBATION THEORY FOR SEMISIMPLE EIGENVALUES APPENDIX 3 II THE STATIONARY PHASE INEQUALITY CHAPTER 4 LINEAR ELLIPTIC GEOMETRIC OPTICS §4 1 EULER’S METHOD AND ELLIPTIC GEOMETRIC OPTICS WITH CONSTANT COE?CIENTS §4 2 ITERATIVE IMPROVEMENT FOR VARIABLE COE?CIENTS AND NONLINEAR PHASES §4 3 FORMAL ASYMPTOTICS APPROACH §4 4 PERTURBATION APPROACH §4 5 ELLIPTIC REGULARITY §4 6 THE MICROLOCAL ELLIPTIC REGULARITY THEOREM CHAPTER 5 LINEAR HYPERBOLIC GEOMETRIC OPTICS §5 1 INTRODUCTION §5 2 SECOND ORDER SCALAR CONSTANT COE?CIENT PRINCIPAL PART 5 2 1 HYPERBOLIC PROBLEMS 5 2 2 THE QUASICLASSICAL LIMIT OF QUANTUM MECHANICS §5 3 SYMMETRIC HYPERBOLIC SYSTEMS §5 4 RAYS AND TRANSPORT 5 4 1 THE SMOOTH VARIETY HYPOTHESIS 5 4 2 TRANSPORT FOR L = L1(?) 5 4 3 ENERGY TRANSPORT WITH VARIABLE COE?CIENTS §5 5 THE LAX PARAMETRIX AND PROPAGATION OF SINGULARITIES 5 5 1 THE LAX PARAMETRIX 5 5 2 OSCILLATORY INTEGRALS AND FOURIER INTEGRAL OPERATORS 5 5 3 SMALL TIME PROPAGATION OF SINGULARITIES 5 5 4 GLOBAL PROPAGATION OF SINGULARITIES §5 6 AN APPLICATION TO STABILIZATION APPENDIX 5 I HAMILTON–JACOBI THEORY FOR THE EIKONAL EQUATION 5 I 1 INTRODUCTION 5 I 2 DETERMINING THE GERM OF ? AT THE INITIAL MANIFOLD 5 I 3 PROPAGATION LAWS FOR ?, D? 5 I 4 THE SYMPLECTIC APPROACH CHAPTER 6 THE NONLINEAR CAUCHY PROBLEM §6 1 INTRODUCTION §6 2 SCHAUDER’S LEMMA AND SOBOLEV EMBEDDING §6 3 BASIC EXISTENCE THEOREM §6 4 MOSER’S INEQUALITY AND THE NATURE OF THE BREAKDOWN §6 5 PERTURBATION THEORY AND SMOOTH DEPENDENCE §6 6 THE CAUCHY PROBLEM FOR QUASILINEAR SYMMETRIC HYPERBOLIC SYSTEMS 6 6 1 EXISTENCE OF SOLUTIONS 6 6 2 EXAMPLES OF BREAKDOWN 6 6 3 DEPENDENCE ON INITIAL DATA §6 7 GLOBAL SMALL SOLUTIONS FOR MAXIMALLY DISPERSIVE NONLINEAR SYSTEMS §6 8 THE SUBCRITICAL NONLINEAR KLEIN–GORDON EQUATION IN THE ENERGY SPACE 6 8 1 INTRODUCTORY REMARKS 6 8 2 THE ORDINARY DI?ERENTIAL EQUATION AND NON- LIPSHITZEAN F 6 8 3 SUBCRITICAL NONLINEARITIES CHAPTER 7 ONE PHASE NONLINEAR GEOMETRIC OPTICS §7 1 AMPLITUDES AND HARMONICS §7 2 ELEMENTARY EXAMPLES OF GENERATION OF HARMONICS §7 3 FORMULATING THE ANSATZ §7 4 EQUATIONS FOR THE PRO?LES §7 5 SOLVING THE PRO?LE EQUATIONS CHAPTER 8 STABILITY FOR ONE PHASE NONLINEAR GEOMETRIC OPTICS §8 1 THE HS ? (RD) NORMS §8 2 HS ? ESTIMATES FOR LINEAR SYMMETRIC HYPERBOLIC SYSTEMS §8 3 JUSTI?CATION OF THE ASYMPTOTIC EXPANSION §8 4 RAYS AND NONLINEAR TRANSPORT CHAPTER 9 RESONANT INTERACTION AND QUASILINEAR SYSTEMS §9 1 INTRODUCTION TO RESONANCE §9 2 THE THREE WAVE INTERACTION PARTIAL DI?ERENTIAL EQUATION §9 3 THE THREE WAVE INTERACTION ORDINARY DI?ERENTIAL EQUATION §9 4 FORMAL ASYMPTOTIC SOLUTIONS FOR RESONANT QUASILINEAR GEOMETRIC OPTICS §9 5 EXISTENCE FOR QUASIPERIODIC PRINCIPAL PRO?LES §9 6 SMALL DIVISORS AND CORRECTORS §9 7 STABILITY AND ACCURACY OF THE APPROXIMATE SOLUTIONS §9 8 SEMILINEAR RESONANT NONLINEAR GEOMETRIC OPTICS CHAPTER 10 EXAMPLES OF RESONANCE IN ONE DIMENSIONAL SPACE §10 1 RESONANCE RELATIONS §10 2 SEMILINEAR EXAMPLES §10 3 QUASILINEAR EXAMPLES CHAPTER 11 DENSE OSCILLATIONS FOR THE COMPRESSIBLE EULER EQUATIONS §11 1 THE 2 ?D ISENTROPIC EULER EQUATIONS §11 2 HOMOGENEOUS OSCILLATIONS AND MANY WAVE INTERACTION SYSTEMS §11 3 LINEAR OSCILLATIONS FOR THE EULER EQUATIONS §11 4 RESONANCE RELATIONS §11 5 INTERACTION COE?CIENTS FOR EULER’S EQUATIONS §11 6 DENSE OSCILLATIONS FOR THE EULER EQUATIONS 11 6 1 THE ALGEBRAIC/GEOMETRIC PART 11 6 2 CONSTRUCTION OF THE PRO?LES