Wave Turbulence Under Parametric Excitation

1 Introduction to Nonlinear Wave Dynamics.- 1.1 Hamiltonian Method for Description of Waves in a Continuous Medium.- 1.1.1 Hamiltonian Equations of Motion.- 1.1.2 Transfer to Complex Variables.- 1.1.3 Hamiltonian Structure Under Small Nonlinearity.- 1.1.4 Dynamic Perturbation Theory. Elimination of “Non-Resonant” Terms from the Hamiltonian.- 1.2 Dimensional Estimation of Hamiltonian Coefficients.- 1.3 Dynamic Equations of Motion for Weakly Non-Conservative Wave Systems.- 1.3.1 Taking into Account Linear Wave Damping.- 1.3.2 Allowing for Thermal Noise.- 1.3.3 Nonlinearity of Wave Damping.- 1.4 Three-Wave Processes.- 1.4.1 Confluence of Two Waves and Other Induced Processes.- 1.4.2 Decay Instability.- 1.4.3 Interaction of Three Waves with Finite Amplitude.- 1.4.4 Explosive Three-Wave Instability.- 1.5 Four-Wave Processes.- 1.5.1 Modulation Instability of the Plane Wave.- 1.5.2 Equation for the Envelopes.- 1.5.3 Package Evolution in Unbounded Media.- 2 The General Properties of Magnetodielectrics.- 2.1 Classification of Substances by Their Magnetic Properties.- 2.1.1 Diamagnets.- 2.1.2 Superconductors.- 2.1.3 Paramagnets.- 2.1.4 Magnetically Ordered Substances (Magnets).- 2.2 Nature of Interaction of Magnetic Moments.- 2.2.1 Exchange Interaction in the Hydrogen Molecule.- 2.2.2 Interatomic Exchange.- 2.2.3 Interatomic Exchange of Large Spins.- 2.2.4 Indirect Exchange Interactions.- 2.2.5 Relativistic Interactions.- 2.3 Energy of Ferromagnets in the Continuum Approximation.- 2.4 Magnetic and Crystallographic Structure of Some Magnets.- 2.4.1 Crystals with Spinel Structure.- 2.4.2 Crystals with Garnet Structure.- 2.4.3 Crystals with Hexagonal Structure.- 2.4.4 Crystals with Rhombohedral Structures.- 3 Spin Waves (Magnons) in Magnetically Ordered Dielectrics.- 3.1 Hamiltonian of Magnons in Ferromagnets (FM).- 3.1.1 Spectrum of Magnons in Cubic Ferromagnets.- 3.1.2 Amplitudes of Three-and Four-Magnon Interaction.- 3.1.3 Three-Magnon Hamiltonian.- 3.1.4 Four-Magnon Interaction Hamiltonian.- 3.2 Hamiltonian Function of Magnons in Antiferromagnets.- 3.2.1 Magnon Spectrum in Antiferromagnets (AFM).- 3.2.2 Interaction Hamiltonian in “Easy-Plane” Antiferromagnets.- 3.2.3 Nuclear Magnons in “Easy-Plane” Antiferromagnets.- 3.3 Comments at the Road Fork.- 3.4 Calculation of Magnon Hamiltonian.- 3.4.1 Equation of Motion of Magnetic Moment.- 3.4.2 Canonical Variables for Spin Waves in Ferromagnets (FM).- 3.4.3 Calculation of Frequencies and Interaction Amplitudes of Waves.- 4 Nonlinear Dynamics of Narrow Packets of Spin Waves.- 4.1 Elementary Processes of Spin Wave Interaction.- 4.1.1 Three-Magnon Processes.- 4.1.2 Modulation Instability of Spin Waves.- 4.2 Self-Focusing of Magnetoelastic Waves in Antiferromagnets (AFM).- 4.2.1 Structure of Basic Equations.- 4.2.2 Properties of Unidimensional Equations.- 4.2.3 Stability of Solitons and Self-Focusing Theorem.- 4.2.4 Evolution of Magnetoelastic Waves in the Absence of a Linear Bond Between Magnons and Phonons.- 4.3 Methods of Parametric Excitation of Spin Waves.- 4.3.1 Transverse Pumping of Spin Waves in FM.- 4.3.2 Parallel Pumping of Spin Waves in FM.- 4.3.3 “Oblique” Pumping of Spin Waves in FM.- 4.3.4 Suhl Instability of the Second Order in FM.- 4.3.5 Parallel Pumping in“Easy-Plane”Antiferromagnets.- 4.3.6 Parametric Pumping of Nuclear Magnons.- 5 Stationary Nonlinear Behavior of Parametrically Excited Waves. Basic S-Theory.- 5.1 History of the Problem.- 5.2 Statement of a Problem of Nonlinear Wave Behavior.- 5.3 Phase Relations and Mechanisms for Amplitude Limitation.- 5.3.1 Analysis of Phase Relations.- 5.3.2 Nonlinear Mechanisms for Limiting Parametric Instability.- 5.4 Basic Equations of Motion in the S-Theory.- 5.4.1 Statistical Properties of a Non-Interacting Field.- 5.4.2 Mean-Field Approximation.- 5.4.3 General Analysis of Basic Equations of S-Theory.- 5.5 Ground State of System of Interacting Parametric Waves.- 5.5.1 Stationary States and Analysis of Instability.- 5.5.2 Ground State Under Low Supercriticality.- 5.5.3 Threshold of Generation of Second Group of Pairs.- 5.5.4 Ground State Under High Supercriticality.- 5.5.5 Nonlinear Susceptibilities of Parametric Waves.- 6 Advanced S-Theory: Supplementary Sections.- 6.1 Ground State Evolution of System with Increasing Pumping Amplitude.- 6.1.1 Ground State of Parametric Waves for Complex Pair Interaction Amplitudes.- 6.1.2 The Second and Intermediate Thresholds.- 6.1.3 Nonlinear Behavior of Non-Analytic Pair Interaction Amplitudes.- 6.2 Influence of Nonlinear Damping on Parametric Excitation.- 6.2.1 Simple Theory.- 6.2.2 Influence of Non-Analyticity on Nonlinear Damping.- 6.3 Parametric Excitation Under the Feedback Effect on Pumping.- 6.3.1 Hamiltonian of the Problem.- 6.3.2 General Analysis of the Equations of Motion.- 6.3.3 First-Order Processes.- 6.3.4 Second-Order Processes.- 6.4 Nonlinear Theory of Parametric Wave Excitation at Finite Temperatures.- 6.4.1 Different Time Correlators and Frequency Spectrum.- 6.4.2 Basic Equations of Temperature S-Theory.- 6.4.3 Separation of Waves into Parametric and Thermal.- 6.4.4 Two-Dimensional Reduction of Basic Equations.- 6.4.5 Distribution of Parametric Waves in k.- 6.4.6 Spectrum of Parametric Waves.- 6.4.7 Heating Below Threshold.- 6.4.8 Influence of Thermal Bath on Total Characteristics.- 6.5 Introduction to Spatially Inhomogeneous S-Theory.- 6.5.1 Basic Equations.- 6.5.2 Parametric Threshold in Inhomogeneous Media.- 6.5.3 Stationary State in Non-Homogeneous Media.- 6.6 Nonlinear Behavior of Parametric Waves from Various Branches. Asymmetrical S-Theory.- 6.6.1 Derivation of Basic Equations.- 6.6.2 Stationary States in Isotropic Case.- 6.7 Parametric Excitation of Waves by Noise Pumping.- 6.7.1 Equations of S-Theory Under Noise Pumping.- 6.7.2 Distribution of Parametric Waves Above Threshold.- 7 Non-Stationary Behavior of Parametrically Excited Waves.- 7.1 Spectrum of Collective Oscillations (CO).- 7.1.1 Spectrum of Spatially Homogeneous CO in the Non-Dissipation Limit.- 7.1.2 Influence of Wave Damping on the CO Spectrum.- 7.1.3 Spectrum of Spatially Non-Homogeneous CO.- 7.2 Linear Theory of CO Resonance Excitation.- 7.2.1 Basic Equations and Their Solution.- 7.2.2 CO Excitation by a Microwave Field.- 7.2.3 Direct CO Excitation by a Radio Frequency Field.- 7.2.4 Coupled Motions of Collective Excitations of Parametric Waves and Sound.- 7.3 Threshold Under Periodic Modulation of Dispersion Law.- 7.4 Large-Amplitude Collective Oscillations and Double Parametric Resonance.- 7.4.1 Stationary State Under Periodic Modulation.- 7.4.2 Parametric Excitation of CO of Parametric Wave System.- 7.5 Transient Processes when Pumping is Turned on.- 7.5.1 Small Supercriticality Range.- 7.5.2 High Supercriticality Range.- 7.6 Parametric Excitation Under Sweeping of Wave Frequency.- 7.6.1 Qualitative Analysis of the Problem.- 7.6.2 Basic Equations of S-Theory Under Frequency Sweeping.- 7.6.3 Solution of S-Theory Equations.- 7.6.4 Dependence of the Number of Waves on the Pumping Amplitude.- 7.7 Problems.- 8 Secondary Parametric Wave Turbulence.- 8.1 Instability of Ground State and Auto-Oscillations.- 8.1.1 Properties and Nature of Spin Wave Oscillations.- 8.1.2 Numerical Simulation of Auto-Oscillation in the S-Theory.- 8.1.3 Conditions for Excitation of Auto-Oscillations.- 8.2 Route to Chaos in Dynamic Systems.- 8.2.1 Introduction.- 8.2.2 Elementary Concepts of Theory of Dynamic Chaos.- 8.2.3 Chaos of Parametric Magnons in CsMnF3.- 8.3 Geometry of Attractors of Secondary Parametric Turbulence of Magnons.- 8.3.1 Effective Phase Space and Dimensionality of Inclusion.- 8.3.2 Experimental Study of Attractor Structure in CsMnF3.- 8.4 Secondary Turbulence and Collapses in Narrow Parametric Wave Packets.- 8.4.1 Equations for Envelopes.- 8.4.2 Stationary Solitons.- 8.4.3 Average Characteristics of Secondary Turbulence.- 8.4.4 Destruction of Parametric Solitons with Large Amplitude.- 8.4.5 Soliton Mechanism of Amplitude Limitation.- 9 Experimental Investigations of Parametrically Excited Magnons.- 9.1 Experimental Investigations of Parametric Instability of Magnons.- 9.1.1 Methods and Materials Investigated.- 9.1.2 Measurements of Constants in Spin Wave Spectra.- 9.1.3 Spin Wave Damping.- 9.2 Nonlinear Behavior of Parametric Magnons — General Information.- 9.2.1 Measuring Technique for Susceptibilities X? and X?.- 9.2.2 Comparison of S-Theory and Experiment for Susceptibilities.- 9.2.3 Measurements of Interaction (Frequency Shift) Amplitude.- 9.2.4 Nonlinear Ferromagnetic Resonance.- 9.3 Investigations of Stationary State With One Group of Pairs.- 9.3.1 Nonlinear Susceptibility in the One-Group State.- 9.3.2 Direct Measurement of Pair Phase.- 9.4 Electromagnetic Radiation of Parametric Magnons.- 9.4.1 Frequency of Parametric Magnons.- 9.4.2 Frequency Width of Parametrically Excited Magnons.- 9.5 Collective Resonance of Parametric Magnons.- 9.5.1 Experimental Technique.- 9.5.2 Frequency of Collective Resonance.- 9.5.3 Susceptibility to Field of Weak Microwave Signal.- 9.5.4 Linewidth of Collective Resonance.- 9.5.5 Oscillations of Longitudinal Magnetization.- 9.5.6 Other Methods for Excitation of Collective Oscillations.- 9.6 Stepwise Excitation in YIG.- 9.6.1 Re-Radiation into the Transverse Channel.- 9.6.2 Interaction of Second-Group Magnons and Transverse Signal.- 9.7 Conditions of Excitation of Auto-Oscillations of Magnons.- 9.7.1 Experimental Setup.- 9.7.2 Intensive Auto-Oscillations of Mode m = 0.- 9.7.3 Crossing the Instability Boundary and Spatially Inhomogeneous Auto-Oscillations.- 9.7.4 Instability of Higher Collective Modes.- 9.8 Effect of Radio-Frequency Field Modulation on Parametric Resonance.- 9.8.1 Suppression of Parametric Instability by Modulation.- 9.8.2 Stationary State of Parametric Magnons Under Modulation of Their Frequency.- 9.9 Double Parametric Resonance and Inhomogeneous Collective Oscillations of Magnons.- 9.10 Parametric Excitation of Magnons Under Noise Modulation of their Frequencies.- 9.10.1 Threshold Amplitude of Noise Pumping.- 9.10.2 Efficiency of Phase Mechanism Under Noise Pumping.- 10 Nonlinear Kinetics of Parametrically Excited Waves.- 10.1 General Equations.- 10.2 Limit of the S-Theory.- 10.2.1 Form of the Green’s Function.- 10.2.2 Separation of the Waves into Parametric and Thermal.- 10.3 Nonlinear Theory of Parametric Excitation of Waves in Random Media.- 10.3.1 General Equations in the S,g2-Approximation.- 10.3.2 Distribution Function of Parametric Waves.- 10.3.3 Behavior of Parametrically Excited Waves Beyond the Threshold.- 10.4 Consistent Nonlinear Theory for Parametric Excitation of Waves.- 10.4.1 Spectral Density of Parametrically Excited Waves.- 10.4.2 Structure of the Distribution Function in k-Space.- References.