Table of Contents
Preface vii
Introduction xv
1 The Solitary Waves Generation due to Passage through the Local Resonance 1
1.1 The Nonlinear Schrödinger Equation. Scattering of Solitons on Resonance 1
1.1.1 Problem Statement and Result 3
1.1.2 Incident Waves 4
1.1.3 Scattering 6
1.1.4 Scattered Waves 8
1.1.5 Numerical Justification of Asymptotic Analysis 10
1.2 Generation of Solitary Packets of Waves in the Nonlinear Klein-Gordon Equation 11
1.2.1 Main Result 11
1.2.2 Pre-Resonance Expansion 14
1.2.3 Internal Asymptotics 16
1.2.4 Post-Resonance Expansion 23
1.3 The Perturbed KDV Equation and Passage through the Resonance 28
1.3.1 Forced Oscillations 29
1.3.2 Inside the Resonance 31
1.3.3 Post-Resonance Expansion 37
1.3.4 Numerical Simulations 39
1.4 Auto-Resonant Soliton and Perturbation with Decaying Amplitude 40
1.4.1 Justification 41
1.4.2 Derivation of the Model Equation for Auto-Resonance 42
1.4.3 An Asymptotic Solution of the Model Equation 43
1.4.4 Effect of Dissipation 46
2 Regular Perturbation of Ill-Posed Problems 48
2.1 Mixed Problems with a Parameter 49
2.1.1 Preliminaries 49
2.1.2 The Cauchy Problem 53
2.1.3 A Perturbation 58
2.1.4 The Main Theorem 62
2.1.5 The Well-Posed Case 65
2.1.6 On Finding the Solution 67
2.1.7 Dirac Operators 72
2.2 Kernel Spikes of Singular Problems 76
2.2.1 Soft Expansions 77
2.2.2 Harmonic Extension 79
2.2.3 Auxiliary Results 80
2.2.4 Formulas for Coefficients 81
2.2.5 Laurent Series 84
2.2.6 Expansion of the Poisson Kernel 85
2.3 An Asymptotic Expansion of the Martinelli-Bochner Integral 86
2.3.1 Asymptotic Expansion 87
2.3.2 The Bochner-Martinelli Integral 88
2.3.3 Regularization 94
2.3.4 Proof of the Theorem 95
2.4 A Formula for the Number of Lattice Points in a Domain 97
2.4.1 Logarithmic Residue Formula 97
2.4.2 The Integral Formula 98
2.4.3 The One-Dimensional Case 101
2.4.4 Some Comments 102
3 Asymptotics at Characteristic Points 103
3.1 Asymptotic Solutions of the 1D Heat Equation 103
3.1.1 Preliminaries 103
3.1.2 On the Heat Equation 104
3.1.3 Blow-Up Techniques 106
3.1.4 Further Reduction 109
3.1.5 The Unperturbed Problem 111
3.1.6 Asymptotic Solutions 113
3.1.7 Local Solvability at a Cusp 117
3.2 EulerTheory on a Spindle 119
3.2.1 Pseudodifferential Operators on Manifolds with Conical Points 120
3.2.2 Merornorphic Families 122
3.2.3 Characteristic Values 123
3.2.4 Factorization 126
3.2.5 Resolvent 129
3.2.6 Unitary Reduction 130
3.2.7 Inhomogeneous Equation 133
3.2.8 Transposed Equations 140
3.2.9 Index 143
3.3 The Laplace-Beltrami Operator on a Rotationally Symmetric Surface 145
3.3.1 Calculus on Singular Varieties 145
3.3.2 Geometry 146
3.3.3 Laplace-Beltrami Operator 148
3.3.4 Weighted Spaces 149
3.3.5 Resolvent 151
3.3.6 Fredholm Theory 153
3.3.7 Asymptotics 155
3.3.8 Index Formula 156
3.4 Boundary Value Problems for Parabolic Equations 158
3.4.1 Anisotropic Ellipticity 161
3.4.2 Parabolicity after Petrovskii 162
3.4.3 Characteristic Points 163
3.4.4 Weighted Spaces 166
3.4.5 Solution in a Special Domain 174
3.4.6 Local Parametrices 182
3.4.7 The Global Parametrix 192
3.4.8 Regularity of Solutions 206
3.4.9 Some Particular Cases 210
4 Asymptotic Expansions of Singular Perturbation Theory 214
4.1 Small Random Perturbations of Dynamical Systems 214
4.1.1 White Noise Perturbation of Dynamical Systems 214
4.1.2 The Case of the Homogeneous Differential Equation 216
4.1.3 The Case of the Homogeneous Boundary Condition 220
4.1.4 The Case of a Right-Hand Side of Zero Average Value 222
4.1.5 Conclusion 222
4.1.6 Appendix 223
4.2 Formal Asymptotic Solutions 225
4.2.1 Asymptotic Phenomena 225
4.2.2 Blow-Up Techniques 228
4.2.3 Formal Asymptotic Solution 230
4.2.4 The Exceptional Case p = 2 233
4.2.5 Degenerate Problem 235
4.2.6 Generalization to Higher Dimensions 235
4.2.7 Parameter-Dependent Norms 240
4.3 The Shapiro-Lopatinskii Condition 241
4.3.1 Boundary Value Problems with Small Parameter 241
4.3.2 Asymptotic Expansion 242
4.3.3 The Main Spaces 245
4.3.4 Auxiliary Results 249
4.3.5 The Main Result 251
4.3.6 Local Estimates in the Interior 253
4.3.7 The Case of Boundary Points 255
4.3.8 Conclusion 257
4.4 Pseudodifferential Calculus with a Small Parameter 257
4.4.1 Singular Problems with a Small Parameter 257
4.4.2 Loss of Initial Data 258
4.4.3 A Passive Approach to Operator-Valued Symbols 260
4.4.4 Operators with a Small Parameter 264
4.4.5 EUipticity with a Large Parameter 269
4.4.6 Another Approach to Parameter-Dependent Theory 269
4.4.7 Regularization of Singularly Perturbed Problems 275
5 Asymptotic Solution of the Schrödinger Equation 278
5.1 Semiclassical Approximations of Quantum Mechanics 278
5.1.1 Standard Approximation 278
5.1.2 Preliminary Results 280
5.1.3 The Schrödinger Equation for Quadratic Hamiltonians 281
5.1.4 Exact Solution with a Rapidly Decreasing Initial Symbol 282
5.1.5 Symmetrized Generating Function 285
5.2 Asymptotic Solution of the Schrödinger Equation 286
5.2.1 Symbol Classes 286
5.2.2 Construction of a Formal Expansion 288
5.2.3 Asymptotic Solution 289
5.2.4 One More Asymptotic Decomposition 291
5.3 The Trace of the Schrödinger Operator 292
5.3.1 Anti-Wick Symbols 292
5.3.2 The Trace Formula 293
5.3.3 Trace Asymptotics for h → 0 293
5.3.4 A Lefschetz Fixed Point Formula 294
5.4 Quantum Dynamics in the Fermi-Pasta-Ulam Problem 295
5.4.1 Wave Decay Processes 296
5.4.2 Classical Limit 297
5.4.3 Quantum Equations of Decay 299
5.4.4 Analysis of Quantum Equations 302
5.4.5 Existence of Solutions 303
5.4.6 Successive Approximations 307
5.4.7 Asymptotic under Large Time 313
5.4.8 Conclusion 314
6 The Kelvin-Helmholtz Instability 316
6.1 Derivation of the Fundamental Equation 318
6.1.1 Setting of the Problem 318
6.1.2 Conditions on the Unknown Boundary 319
6.1.3 Derivation of an Equation for the Curve 320
6.1.4 A Hamiltonian Form of the Equation of Tangential Discontinuity 322
6.1.5 Conservation Laws 323
6.2 Small Perturbation of Tangential Discontinuity 324
6.2.1 Linearization of the Equation of Tangential Discontinuity 324
6.2.2 On the Ellipticity of the System (6.18) 326
6.2.3 Small Perturbations of Rectilinear Tangential Discontinuity 327
6.2.4 The Linearization of Equation (6.17) 328
6.2.5 Remarks on the Linearized System 329
6.3 Analytic Continuation from a Boundary Subset 331
6.3.1 The Riemann Mapping Theorem 333
6.3.2 Hardy Spaces 334
6.3.3 The Cauchy Formula 335
6.3.4 Quenching Functions 336
6.3.5 The Goluzin-Krylov Formula 338
6.3.6 A Uniqueness Theorem 339
6.3.7 Approximation through Holomorphic Functions 340
6.3.8 Expansion in a Fourier Series 341
6.3.9 Approximation through Legendre Polynomials 343
6.4 A Numerical Approach to the Riemann Hypothesis 344
6.4.1 The Riemann Zeta Function 345
6.4.2 Analytic Continuation in a Lune 346
6.4.3 A Carleman Formula for a Half-Disk 349
6.4.4 Reduction of the Riemann Hypothesis 352
6.4.5 Numerical Experiments 355
7 Nonlinear Cauchy Problems for Elliptic Equations 357
7.1 A Variational Approach to the Cauchy Problem 357
7.1.1 Relaxations of Ill-Posed Problems 357
7.1.2 The Cauchy Problem 359
7.1.3 Variational Setting 361
7.1.4 Euler's Equations 364
7.1.5 Examples 366
7.1.6 Mixed Problems 367
7.1.7 Inverse Problem Approach 369
7.2 The Cauchy Problem for Chaplygin's System 371
7.2.1 Preliminaries 371
7.2.2 Chaplygin's System 372
7.2.3 Variational Setting 373
7.2.4 Existence of Solutions 375
7.2.5 Stable Cauchy Problems 377
7.2.6 Approximate Solutions 379
7.3 Hyperbolic Formulas in Elliptic Cauchy Problems 380
7.3.1 The Cauchy Problem 384
7.3.2 Hyperbolic Reduction 385
7.33 The Planar Case 387
7.3.4 The Carleman Formula 389
7.3.5 Poisson's Formula 391
7.3.6 The Kirchhoff Formula 393
7.3.7 Concluding Remarks 395
7.4 A WKB Solution to the Navier-Stokes Equations 396
7.4.1 Basic Equations of the Dynamics of an Incompressible Viscous Fluid 396
7.4.2 Generalized Navier-Stokes Equations 398
7.4.3 Energy Estimates 401
7.4.4 First Steps towards the Solution 403
7.4.5 A WKB Solution 405
Bibliography 407
Index 421
