Mathematics of Wave Phenomena

Preface
Contents
Morawetz Inequalities for Water Waves
1 Introduction
2 Water Waves
3 Morawetz Estimates
4 Morawetz Estimates for Two Dimensional Water Waves
5 Holomorphic Coordinates
6 Further Questions
References
Numerical Study of Galerkin–Collocation Approximation in Time for the Wave Equation
1 Introduction
2 Mathematical Problem and Notation
3 Galerkin–Collocation Schemes
4 Galerkin–Collocation GCC1(3)
4.1 Fully Discrete System
4.2 Solver Technology
4.2.1 First Approach: Condensing the Linear System
4.2.2 Second Approach: Solving the Non-symmetric System
4.3 Numerical Convergence Tests
4.4 Test Case of Structural Health Monitoring
5 Galerkin–Collocation GCC2(5)
5.1 Fully Discrete System
5.2 Iterative Solver and Convergence Study
References
Effective Numerical Simulation of the Klein–Gordon–Zakharov System in the Zakharov Limit
1 Introduction
2 From the KGZ System to the Zakharov System
3 Error Bounds for the Numerical Scheme
4 Some Numerical Illustrations
4.1 Efficiency
4.2 Asymptotic Consistency Plot
References
Exponential Dichotomies for Elliptic PDE on Radial Domains
1 Introduction
1.1 Outline of the Paper
2 Construction of the Exponential Dichotomy
2.1 The Limiting Operator
2.2 The Perturbation
2.3 Unique Continuation
2.4 Proof of Theorem 2 and Corollary 1
3 Dichotomy Subspaces and Spherical Harmonics
3.1 The Dichotomy Subspaces
3.2 The Dichotomy Projections
3.3 The Evolution Operators
3.4 Liouville-Type Theorems
4 Applications
4.1 Eigenvalue Problems
4.2 Reformulation of Two Nonlinear Problems
4.2.1 A Nonlinear Boundary Value Problem
4.2.2 A Nonlinear Problem on Rn
References
Stability of Slow Blow-Up Solutions for the Critical Focussing Nonlinear Wave Equation on R3+1
1 Introduction
2 The Construction of Slow Blow-Up Solutions
2.1 The Renormalization Step
2.2 Completion to an Exact Solution
3 The Stability of Slow Blow-Up Solutions
3.1 Conditional Stability Result
3.2 Optimal Stability Result
References
Local Well-Posedness for the Nonlinear Schrödinger Equation in the Intersection of Modulation Spaces Mp, qs(Rd) M∞, 1(Rd)
1 Introduction
Notation
2 Preliminaries
3 Littlewood–Paley Theory
4 Algebra Property and Hölder-Type Inequality
5 Proof of the Local Well-Posedness, Theorem 4
References
FEM-BEM Coupling of Wave-Type Equations: From the Acoustic to the Elastic Wave Equation
1 Introduction and Motivation
2 Problem Statement and Background
3 Calderón Operator
4 Space Discretization
5 Time-Discretization
6 Stability Analysis
7 Conclusion
References
On Hyperbolic Initial-Boundary Value Problems with a Strictly Dissipative Boundary Condition
1 Introduction and Main Result
2 Proof of Theorem 1
3 Maxwell's Equation
4 More on Strictly Dissipative Boundary Operators
Appendix
References
On the Spectral Stability of Standing Waves of Nonlocal PT Symmetric Systems
1 Introduction
2 Nonlocal NLS Models
2.1 Nonlocal Space NLS Model
2.1.1 The Eigenvalue Problem for the Nonlocal Space NLS Model
2.1.2 Stability Analysis of the Waves
2.2 Nonlocal Time NLS and Nonlocal in Space and Time NLS Models
2.2.1 The Eigenvalue Problem for Nonlocal Time NLS Model
2.2.2 The Eigenvalue Problem of the Nonlocal in Space and Time NLS Models
3 Klein–Gordon Models
3.1 Reverse Time Nonlocal KG Equation
3.2 Reverse Space-Time Nonlocal KG Equation
References
Sparse Regularization of Inverse Problems by Operator-Adapted Frame Thresholding
1 Introduction
2 Diagonal Frame Decomposition
2.1 Formal Definition
2.2 Radon Transform
2.3 Inversion of the Wave Equation
3 Sparse 1-Regularization
3.1 1-Analysis Regularization
3.2 Synthesis Regularization
3.3 Sparse Regularization Using an SVD
4 Regularization via DFD Thresholding
4.1 DFD Soft-Thresholding
4.2 Convergence Analysis
4.3 Convergence Rates
5 Conclusion
References
Soliton Solutions for the Lugiato–Lefever Equation by Analytical and Numerical Continuation Methods
1 Introduction
2 Existence of Bright Solitons on the Real Line
3 Numerical Experiments
3.1 Continuation in f̃ and
3.2 Rescaling Back to (2) and Extending by a Constant
3.3 An Example
3.4 Scanning Large Ranges of the Forcing and Detuning Variable
3.5 Border Curves in the ζ-f Stability Chart
References
Error Analysis of Discontinuous Galerkin Discretizations of a Class of Linear Wave-type Problems
1 Introduction
2 Notation
3 Analytical Properties of Friedrichs' Systems
4 Spatial Discretization
4.1 Discrete Setting
4.2 Friedrichs' Operators in the Discrete Setting
4.3 Discrete Friedrichs' Operators
4.4 Spatial Discretization of the Wave-type Problem
5 Error Analysis of the Spatially Semi-discrete Problem
6 Concluding Remarks
Appendix: Proofs from Sect.4
References
Ill-posedness of the Third Order NLS with Raman Scattering Term in Gevrey Spaces
1 Introduction
2 Reduction to ODE
3 Multilinear Estimates
4 Proof of Theorem 1
Appendix: Topology on Gσ
References
Invariant Measures for the DNLS Equation
1 Introduction
2 Main Results
2.1 Strategy of the Proof
References
A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains
1 Introduction and Main Results
2 Notations, Preliminaries, and Proofs
3 Generalizations and the Classical Div-curl-Lemma
References
Existence and Stability of Klein–Gordon Breathers in the Small-Amplitude Limit
1 Introduction
2 Existence via Lyapunov–Schmidt Decomposition
3 Stability via Lyapunov–Schmidt Decomposition
4 Long-Time Nonlinear Stability via Resonant Normal Forms
4.1 Setting, Preliminaries and Normal Form Result
4.2 High Order Approximation and Nonlinear Stability Results
4.3 Proof of Theorem 4 (Normal Form Theorem)
4.4 Proof of Theorem 5 (High Order Approximation)
4.5 Proof of Theorem 6 (Exponentially Long Time Stability)
References
On Strichartz Estimates from 2-Decoupling and Applications
1 Introduction
2 Linear Strichartz Estimates
3 Bilinear Strichartz Estimates and Transversality
4 Local Well-Posedness of Generalized Cubic Schrödinger Equation
References
On a Limiting Absorption Principle for Sesquilinear Forms with an Application to the Helmholtz Equation in a Waveguide
1 Introduction
2 Fredholm Alternative and Limiting Absorption Principle
2.1 Fredholm Alternative for Sesquilinear Forms
2.2 Limiting Absorption Principle for Sesquilinear Forms
3 Existence Result for a Periodic Bounded Waveguide
3.1 Setting
3.2 Existence Result for the Helmholtz Equation with Damping
3.3 Existence Result for the Helmholtz Equation
References
Some Inverse Scattering Problems for Perturbations of the Biharmonic Operator
1 Introduction
2 Solvability of Direct Scattering Problems
3 Proof of the Main Results
4 Conclusions
References