Nonlinear Optical and Atomic Systems At the Interface of Physics and Mathematics

Focusing on the interface between mathematics and physics, this book offers an introduction to the physics, the mathematics, and the numerical simulation of nonlinear systems in optics and atomic physics. The text covers a wide spectrum of current research on the subject, which is an extremely activ...

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Main Authors Besse, Christophe, Garreau, Jean-Claude
Format eBook Book
LanguageEnglish
Published Cham Springer Nature 2015
Springer
Springer International Publishing AG
Springer International Publishing
Springer International Publisher
Edition1
SeriesLecture Notes in Mathematics
Subjects
Atomic, Molecular, Optical and Plasma Physics
Atoms
Chaotic Dynamics
Condensed Matter
Disordered Systems and Neural Networks
Dynamical Systems
Gross-Pitaevskii equations
Mathematical Physics
Mathematics
Mathematics and Statistics
Nonlinear optics
Nonlinear Sciences
Physical optics
Physics
Quantum Physics
Online AccessGet full text
ISBN3319190156
9783319190150
3319190148
9783319190143
ISSN0075-8434
1617-9692
DOI10.1007/978-3-319-19015-0

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Abstract Focusing on the interface between mathematics and physics, this book offers an introduction to the physics, the mathematics, and the numerical simulation of nonlinear systems in optics and atomic physics. The text covers a wide spectrum of current research on the subject, which is an extremely active field in physics and mathematical physics, with a very broad range of implications, both for fundamental science and technological applications: light propagation in microstructured optical fibers, Bose-Einstein condensates, disordered systems, and the newly emerging field of nonlinear quantum mechanics. Accessible to PhD students, this book will also be of interest to post-doctoral researchers and seasoned academics.
AbstractList Focusing on the interface between mathematics and physics, this book offers an introduction to the physics, the mathematics, and the numerical simulation of nonlinear systems in optics and atomic physics. The text covers a wide spectrum of current research on the subject, which is an extremely active field in physics and mathematical physics, with a very broad range of implications, both for fundamental science and technological applications: light propagation in microstructured optical fibers, Bose-Einstein condensates, disordered systems, and the newly emerging field of nonlinear quantum mechanics. Accessible to PhD students, this book will also be of interest to post-doctoral researchers and seasoned academics.
Author Garreau, Jean-Claude
Besse, Christophe
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Notes CEMPI Centre Européen pour les Mathématiques, la Physique et leurs interactions
This is the first volume of the new "CEMPI subseries"--p. v
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Snippet Focusing on the interface between mathematics and physics, this book offers an introduction to the physics, the mathematics, and the numerical simulation of...
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SubjectTerms Atomic, Molecular, Optical and Plasma Physics
Atoms
Chaotic Dynamics
Condensed Matter
Disordered Systems and Neural Networks
Dynamical Systems
Gross-Pitaevskii equations
Mathematical Physics
Mathematics
Mathematics and Statistics
Nonlinear optics
Nonlinear Sciences
Physical optics
Physics
Quantum Physics
Subtitle At the Interface of Physics and Mathematics
TableOfContents 4.6.1 Experiment I: Dynamics of a Rotating BEC in a Harmonic Trap -- 4.6.2 Experiment II: Dynamics of a BEC in Quadratic-Plus-Quartic Trap -- 4.6.3 Experiment III: Dynamics of a 2d Dark Soliton -- 5 Computation of the Dynamics with Stochastic Terms -- 5.1 Numerical Schemes for the Stochastic GPE -- 5.1.1 The Time-Splitting Schemes -- 5.1.2 The Relaxation Scheme -- 5.2 Numerical Examples -- 5.2.1 Simulation of Fractional Brownian Motions -- 5.2.2 Order in Time of the Schemes for the Stochastic GPE -- 6 Conclusion -- References -- Orbital Stability: Analysis Meets Geometry -- 1 Introduction -- 1.1 Notions of Stability -- 1.2 Symmetries and Relative Equilibria -- 1.3 Examples -- 2 Dynamical Systems, Symmetries and Relative Equilibria -- 2.1 Dynamical Systems on Banach Spaces -- 2.2 Symmetries, Reduced Dynamics and Relative Equilibria -- 3 Examples -- 3.1 Motion in a Spherical Potential -- 3.2 The Nonlinear Schrödinger Equation -- 3.3 The Manakov Equation -- 3.4 The Nonlinear Wave Equation -- 3.5 Generalized Symmetries -- 4 Orbital Stability: A General Definition -- 5 Orbital Stability in Spherical Potentials -- 5.1 Fixed Points -- 5.2 Circular Orbits -- 6 Hamiltonian Dynamics in Infinite Dimension -- 6.1 Symplectors, Symplectic Banach Triples, Symplectic Transformations, Hamiltonian Vector Fields -- 6.2 Hamiltonian Flows and Constants of the Motion -- 6.3 Symmetries and Noether's Theorem -- 6.4 Linear Symplectic Flows -- 6.5 Hamiltonian PDE's: Examples -- 7 Identifying Relative Equilibria -- 8 Orbital Stability: An Abstract Proof -- 8.1 Introduction: Strategy -- 8.2 A Simple Case -- 8.3 Coercivity Implies Stability I -- 8.4 Sufficient Condition for Coercivity -- 8.5 Coercivity Implies Stability II -- 9 Plane Wave Stability on the Torus for NLS -- 9.1 Orbital Stability -- 10 Orbital Stability for Inhomogeneous NLS -- 10.1 Hamiltonian Setting
10.2 Bifurcation Results -- 10.3 Stability -- 10.3.1 Local Stability by Bifurcation -- 10.3.2 Global Continuation -- 11 A Brief History of Orbital Stability -- Appendix -- Differential Geometry: The Basics -- Lie Algebras, Lie Groups and Their Actions -- Hamiltonian Dynamical System with Symmetry in Finite Dimension -- Hamiltonian Dynamical Systems -- Symmetries and Constants of the Motion -- References -- High-Frequency Dynamics for the Schrödinger Equation, with Applications to Dispersion and Observability -- 1 Introduction -- 1.1 The Schrödinger Equation -- 1.2 Some (More or Less) Explicit Examples -- 1.3 A First Approach to the Study of the Dynamics: The Correspondence Principle and the Semiclassical Regime -- 1.4 Semiclassical Analysis of the Non-semiclassical Problem -- 2 The Compactness Approach to the Study of the Dynamics of the Schrödinger Flow -- 2.1 Description of the Problem -- 2.2 Dispersive Effects -- 2.3 Unique Continuation and Observability -- 3 Pseudodifferential Operators and Semiclassical Measures -- 3.1 Basic Notions from the Theory of Pseudodifferential Operators -- 3.2 Symbolic Calculus for Semiclassical Pseudodifferential Operators -- 3.3 Operators on a Manifold -- 3.4 Semiclassical Measures: Motivations -- 3.5 Semiclassical Measures, Definition -- 3.6 Semiclassical Measures, Properties and Examples -- 4 Semiclassical Measures and the Schrödinger Flow -- 4.1 Semiclassical Propagation and Egorov's Theorem -- 4.2 The Ehrenfest Time -- 4.3 Beyond the Eherenfest Time -- 4.4 Concentration of Laplacian Eigenfunctions -- 5 Results in Completely Integrable Geometries -- 5.1 Averaging and Zoll Manifolds -- 5.2 Flat Tori -- 5.3 Some Ideas from the Proof of Theorem 7 -- 5.4 Other Results in Other Geometries -- References -- Index
Intro -- Preface -- Acknowledgements -- Contents -- List of Contributors -- Nonlinear Lattice Waves in Random Potentials -- 1 Introduction -- 2 Lattice Wave Equations -- 3 Anderson Localization -- 4 Adding Nonlinearity -- 4.1 The Secular Normal Form -- 4.2 Expected Dynamical Regimes -- 4.3 Beyond the Secular Normal Form -- 4.4 Measuring Properties of Spreading Wave Packets -- 5 Computing Spreading Wave Packets: Collecting Evidence -- 5.1 Single Site Excitations -- 5.2 Single Mode Excitations -- 5.3 Normal Mode Dephasing -- 6 Nonlinear Diffusion -- 6.1 Measuring Chaos -- 6.2 Effective Noise Theory -- 6.3 Generalizations -- 7 Testing the Predictions -- 7.1 The Crossover from Strong to Weak Chaos -- 7.2 Density Profile Scaling -- 7.3 Tuning the Power of Nonlinearity and the Lattice Dimension -- 7.4 Heat Conductivity -- 7.5 Ramping Nonlinearity -- 8 Correlated Potentials -- 8.1 Subdiffusive Destruction of Aubry-Andre Localization -- 8.2 Subdiffusive Destruction of Wannier-Stark Localization -- 8.3 Subdiffusive Destruction of Dynamical Localization -- 9 Discussion -- References -- Modeling and Computation of Bose-Einstein Condensates: Stationary States, Nucleation, Dynamics, Stochasticity -- 1 Modeling: Bose, Einstein, Gross, and Pitaevskii -- 1.1 From the Theory to the Realization of Bose-Einstein Condensates -- 1.2 Modeling Bose-Einstein Condensates -- 1.2.1 From Classical to Quantum Mechanics -- 1.2.2 Application to Bose-Einstein Condensates -- 1.3 Enrichment of the GPE: Quantum Vorticity, Dipole-Dipole Interaction, Multi-components, Stochasticity -- 1.3.1 Rotating Bose-Einstein Condensates and Quantum Vortices -- 1.3.2 BECs Including Dipolar Interactions -- 1.3.3 Origin of Stochastic Effects in BECs -- 1.3.4 Multi-components BECs -- 1.4 Stationary States -- 1.4.1 Critical Points of the Energy Functional E -- 1.4.2 Ansatz of the Stationary States
1.5 The Rotating GPE with a Quadratic Potential: Dimensionless Form in 3d, 2d and 1d -- 1.5.1 Dimensionless Form of the GPE -- 1.5.2 Dimension Reductions -- 2 Stationary States and Nucleation of Quantized Vortices -- 2.1 Stationary States Formulation: Solving a Minimization Problem for the Energy Functional or a Nonlinear Eigenvalue Problem (Under Constraint)? -- 2.2 Time and Space Discretizations of System (24) -- 2.2.1 Semi-implicit Backward Euler Scheme in Time -- 2.2.2 Spatial Discretizations -- 2.2.3 Fully Discretized Semi-implicit Crank-Nicolson Scheme -- 2.2.4 BESP or CNSP? That Is the Question -- 2.2.5 BESP or BEFD? This Is Another Question -- 2.3 Which Initial Guess for CNGF? -- 2.4 Limitation of the Fixed Point Method for BESP -- 2.5 Extension to Multi-components BECs -- 2.5.1 CNGF for Multi-components BECs -- 2.5.2 BESP for Multi-components BECs -- 3 The Gross-Pitaevskii Equation Laboratory -- 3.1 GPELab: A Short Presentation -- 3.2 Experiment I: Stationary State of a 1d BEC with Josephson Junction -- 3.3 Experiment II: Stationary State of a Fast Rotating 2d BEC in a Strongly Confining Trap -- 3.4 Experiment III: Stationary State of a 3d Dipole-Dipole BEC -- 4 Computation of the Dynamics -- 4.1 Dynamics of the GPE and Continuous/Discrete Properties -- 4.2 Time-Splitting Pseudo-spectral Schemes for the Rotating GPE -- 4.2.1 General Principle of Time-Splitting Techniques -- 4.2.2 Lie Time-Splitting Scheme for (87) -- 4.2.3 Strang Time-Splitting Scheme for (87) -- 4.3 The Relaxation Scheme for the Rotating GPE -- 4.4 Other Schemes: Euler, Crank-Nicolson, Leap-Frog, Rotating Frame System -- 4.5 The Multi-components Case -- 4.5.1 Time-Splitting Schemes for a System with Nc Components -- 4.5.2 Relaxation Scheme for a System with Nc Components -- 4.6 Numerical Study of the TSSP1-ADI, TSSP2-ADI and ReSP Schemes for the Dynamics of Rotating GPEs
Title Nonlinear Optical and Atomic Systems
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