The Nonlinear Schrodinger Equation

Transcription

1 Catherine Sulem Pierre-Louis Sulem The Nonlinear Schrodinger Equation Self-Focusing and Wave Collapse Springer

2 Preface v I Basic Framework 1 1 The Physical Context Weakly Nonlinear Dispersive Waves A Weakly Nonlinear Dispersion Relation Derivation in Terms of Fourier-Mode Coupling Introduction to Multiple-Scale Analysis The Example of Optical Waves Waves in a Dielectric The Paraxial Approximation in a Linear Medium Static Modulation in a Kerr Medium Time Dispersion of Ultrashort Wave Trains Basic Dynamical Effects Modulational Instability Solitons in One Space Dimension Soliton Instability for Transverse Perturbations Fluid-Dynamical Form of the NLS Equation 25 2 Structural Properties Variational Formulation Lagrangian Structure Hamiltonian Structure 29

3 xii 2.2 The Noether Theorem Invariances and Conservation Laws Variance Identity Elliptic NLS Equation Extension to the Non-Elliptic NLS Equation II Rigorous Theory 41 3 Existence and Long-Time Behavior The Linear Schrodinger Operator Basic Estimates Linear Schrodinger Equation with Potential Smoothing Properties Existence Properties Local Existence Global Existence for Large Initial Conditions Global Regularity for Small Initial Conditions Self-Similar Solutions Scattering Properties The Case of Repulsive Nonlinearity The case of Attracting Nonlinearity Further Results Generalized NLS Equations Defocusing NLS with Nonzero Condition at Infinity Periodic Boundary Conditions and Invariant Measures 68 4 Standing Wave Solutions A Heuristic Approach Linear Stability Analysis The Case of Finite Amplitude Perturbations Existence and Variational Approach Necessary Conditions for Existence in H Existence Results Variational Approaches Stability/Instability Conditions Subcritical Dimension: Orbital Stability Critical Case: Instability by Blowup Supercritical Case: Instability by Blowup A General Approach for Hamiltonian Systems Setting of the Problem and Main Results Main Steps of the Proofs Extension to Abstract Hamiltonian Systems... 90

4 xiii 4.5 Further Stability Results Asymptotic Stability Results Ground-State Orbital Stability and Global Existence 92 5 Blowup Solutions Finite-Time Blowup Case of Finite Variance Extensions to Solutions with Infinite Variance Analysis of the Blowup Rate of Blowup A Self-Similar Solution at Critical Dimension Solutions with Exactly k Blowup Points X 2 -Norm Concentration at Critical Dimension III Asymptotic Analysis near Collapse Numerical Observations Capturing the Blowup Structure A Scale Transformation for Isotropic Solutions Dynamic Rescaling Simulation of Isotropic Collapse Stability of Supercritical Self-Similar Solutions The NLS Equation at Critical Dimension An Adaptive Galerkin Finite Element Method Simulation of Non-Radially Symmetric Solutions Anisotropic Dynamic Rescaling Stability of Isotropic Collapse An Iterative Grid Redistribution Method Supercritical Collapse Self-Similar Blowup Solutions Properties of the Profile Spatial Extension of the Self-Similar Profile Rate of Convergence to Self-Similar Solutions Dissipation and Postcollapse Dynamics Critical Collapse Self-Similar Profile near Critical Dimension Constraints on the Self-Similar Profile A Nonlinear"Eigenvalue Problem A Nonuniform Limit Remarks on the Critical Profile Asymptotic Solutions at Critical Dimension 151

5 xiv Construction of Asymptotic Solutions Effects of Mass and Hamiltonian Radiation Adiabatic Approximation and Blowup Time Estimate Relation with Standing-Wave Instability Perturbations of Focusing NLS Wave Dissipation at Critical Dimension Effect of an Individual Collapse The Turbulent Regime Non-Elliptic Schrodinger Equation Estimates for the Partial Variances Numerical Observations in Two Dimensions Numerical Simulations in Three Dimensions Effect of a Small Time Dispersion on Critical Collapse Saturated Nonlinearity Standing-Wave Solutions Numerical Observations Saturation of Critical Collapse Other Pertubations Weak Nonparaxiality General Formalism 185 IV Coupling to a Mean Field Mean Field Generation General Formalism A few simple examples The Korteweg-de Vries equation The Boussinesq equation The Kadomtsev-Petviashvili equation A Degenerate Case Gravity-Capillary Surface Waves The Water-Wave Problem Equations Governing the Interface Motion Formal Modulation Analysis Error Bounds Preliminaries Modulation of the Water-Wave Operator The Davey-Stewartson System General Setting 221

6 xv Boundary Conditions Expression of the Mean Flow Conservation Properties Standing-Wave Solutions The Elliptic-Elliptic Case The Hyperbolic-Elliptic case The Initial Value Problem Subsonic Wave Packet Supersonic Wave Packets Rate of Blowup for Elliptic-Elliptic DS Solutions of Elliptic-Hyperbolic DS 240 V Coupling to Acoustic Waves Langmuir Oscillations Derivation of the Zakharov Equations The Two-Fluid Model The Vector Zakharov Equations The Electrostatic Limit Generation of a Large-Scale Magnetic Field Rigorous Results Existence Theory The Subsonic Limit The Vector NLS Equation Evidence of Collapse Heuristic Arguments Simulations in the Electrostatic Approximation Simulations of the Vector Equations The Scalar Model Self-Similar Solutions Formal Construction Dynamical Stability Existence and Blowup Results Existence Theory Blowup Results Further Analysis in Two Dimensions Existence of Self-Similar Blowup Solutions Mass Concentration Properties A Sharp Existence Result Instability of Standing Wave Solutions An Optimal Lower Bound for the Blowup Rate.. 283

7 xvi 15 Progressive Waves in Plasmas Interaction with a Nonmagnetic Medium Laser Beams in Plasmas Hamiltonian Formalism Alfven Wave Filamentation Modulation Equations Influence of the Magnetosonic Waves Weakly Dispersive Alfven Waves Parallel Propagation Oblique Propagation 306 Synopsis 307 References 309 Name Index 339 Subject Index 347