The beauty of the theoretical science is that quite different physical, biological, etc. phenomena can often be described as similar mathematical objects, by similar differential (or other) equations. In the 20th century, the notion of ""theory of oscillations"" and later ""theory of waves"" as unifying concepts, meaning the application of similar methods and equations to quite different physical problems, came into being. In the variety of applications (quite possibly in most of them), the oscillatory process is characterized by a slow (as compared with the characteristic period) variation of its parameters, such as the amplitude and frequency. The same is true for the wave processes. This book describes a variety of problems associated with oscillations and waves with slowly varying parameters. Among them the nonlinear and parametric resonances, self-synchronization, attenuated and amplified solitons, self-focusing and self-modulation, and reaction-diffusion systems. For oscillators, the physical examples include the van der Pol oscillator and a pendulum, models of a laser. For waves, examples are taken from oceanography, nonlinear optics, acoustics, and biophysics. The last chapter of the book describes more formal asymptotic perturbation schemes for the classes of oscillators and waves considered in all preceding chapters.
Author(s): Lev Ostrovsky
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 371
City: Singapore
Contents
Preface
Introduction
References
Chapter 1 Perturbed Oscillations
1.1 Linear Oscillator with Damping
1.2 Oscillator with Cubic Nonlinearity
1.3 Oscillator Under the Action of External Force. Resonance
1.4 A Forced Nonlinear Oscillator
1.5 Oscillators with Variable Parameters. Parametric Resonance
1.5.1 Slowly varying parameters. WKB approximation
1.5.2 Parametric resonance
1.6 Active Systems. The van der Pol Oscillator
1.7 A Lumped Model of Laser
1.8 Strongly Nonlinear Oscillators. A Pendulum
1.8.1 Ideal pendulum
1.8.2 Damping oscillations
1.9 A Charged Particle in the Magnetic Field
1.10 Interaction of Nonlinear Oscillators
1.11 Synchronization
1.11.1 Coupled Duffing oscillators
1.11.2 Synchronization of active oscillators
1.12 Self-Synchronization in Ensembles of Oscillators
1.12.1 Synchronization of limit cycles. Kuramoto model.
1.12.2 Auto-synchronization of Duffing oscillators
1.13 Variable-Parameter Chaotic Oscillations
Appendix A. The Jacobi Elliptic Functions
Appendix B. Phase Plane
References
Chapter 2 Linear Waves
2.1 Kinematics of Waves. Phase and Group Velocity
2.2 Klein-Gordon Equation with Dissipation
2.2.1 Non-dissipative KG equation
2.2.2 KG with dissipation
2.3 Linear Schrödinger Equation
2.3.1 General form
2.3.2 Gaussian impulse
2.4 Evolution of Wave Amplitude and Wavenumber
2.4.1 General equations
2.4.2 Self-similar solutions
2.4.3 Fresnel integrals
2.5 Asymptotic Behavior of Linear Waves
2.5.1 Method of stationary phase
2.5.2 Airy function
2.6 Wave Beams
2.6.1 Monochromatic beams
2.6.2 Space-time beams
2.7 Frequency-Modulated Dispersive Waves: Compression and Spreading
2.7.1 Space-time rays
2.7.2 Variation of wave energy and amplitude
2.7.3 Asymptotic of the envelope waves
2.8 Example: Water Waves
2.8.1 Dispersion relation
2.8.2 Deep-water waves
2.8.3 Shallow-water waves
2.9 Geometrical Theory of Waves
2.9.1 General relations
2.9.2 Geometrical acoustics
2.9.3 One-dimensional propagation. Waves in the atmosphere
2.10 Waves in Media with Time-Variable Parameters
2.10.1 Media with traveling-wave parameters
2.10.2 Trapping and blocking wave packets
References
Chapter 3 Nonlinear Quasi-Harmonic Waves
3.1 Nonlinear Schrödinger Equation
3.1.1 General form
3.1.2 Nonlinear Klein-Gordon equation
3.1.3 Nonlinear electromagnetic waves in a dispersive dielectric
3.2 Nonlinear Waves of Envelopes
3.2.1 Variation of the wave amplitude and wavenumber
3.2.2 Modulation instability
3.2.3 Simple envelope waves
3.3 Stationary Envelope Waves
3.3.1 “Bright” and “dark” envelope solitons
3.3.2 Envelope waves with frequency modulation
3.3.3 Envelope shock waves
3.4 Stationary Beams. Self-Focusing
3.4.1 Nonlinear parabolic equation
3.4.2 Instability of plane wave
3.4.3 A localized nonlinear beam
3.4.4 Self-similar nonlinear beams
3.4.5 Strong self-focusing
3.5 Space-Time Effects
3.6 The Gross-Pitaevskii Equation
3.7 Resonant Interactions of Waves
3.7.1 Resonant triplet
3.7.2 Frequency doubling and period doubling
3.7.3 Non-dispersive media. “Parametric” arrays
References
Chapter 4 Modulated Non-Sinusoidal Waves
4.1 Strong Nonlinear Klein-Gordon(KG) Equation
4.1.1 Averaged equations
4.1.2 KG with cubic nonlinearity
4.1.3 Modulation of a periodic wave
4.1.4 Attenuation of nonlinear wave
4.2 Korteweg-de Vries Equation
4.2.1 Stationary progressive waves
4.2.2 Attenuation of cnoidal waves
4.2.3 Waves in the inhomogeneous media. Water waves over a sloping bottom
4.3 Conservation Equations and Evolution of Stepwise Function
4.3.1 Whitham’s theory
4.3.2 Evolution of step function
References
Chapter 5 Slowly Varying Solitons
5.1 Perturbed KdV Equation
5.1.1 The equation for the soliton amplitude
5.1.2 KdV equation with dissipation
5.1.3 Radiation from a soliton
5.2 The Nonlinear Klein-Gordon Equation
5.3 Nonlinear Shrödinger Equation (NSE)
5.4 Rotational KdV Equation
5.4.1 Terminal damping of solitons
5.4.2 Radiation
5.5 Refraction of Solitons
5.5.1 Geometrical theory of solitons
5.5.2 Transverse stability of a soliton
5.5.3 Circular fronts Self-refraction of solitons
5.6 Damping of 2D Solitons in the Kadomtsev-Petviashvili (KP) Equation
5.6.1 Two-dimensional solitons
5.6.2 Damping of a lump
References
Chapter 6 Interactions of Solitons, Kinks, and Vortices
6.1 Types of Soliton Interactions: Repulsion, Attraction and the Bound States
6.2 A Generalized KdV Equation
6.2.1 Lagrangian approach
6.2.2 Interaction of the KdV solitons
6.2.3 The Kawahara equation
6.3 Soliton Lattices
6.3.1 Soliton-soliton structures and hierarchy
6.3.2 Stable and unstable soliton structures
6.4 Interaction of Solitons in Electromagnetic Lines
6.5 Interaction of Kinks and Flat-Top Solitons
6.5.1 The sine-Gordon equation
6.5.2 Attenuation of SG kinks
6.5.3 Interaction of SG kinks
6.5.4 Compound solitons in the Gardner equation
6.5.5 Two-soliton interaction
6.5.6 Modulated lattices of kinks
6.5.7 Large-amplitude internal waves in the ocean
6.6 Two- and Three-Dimensional Solitons
6.6.1 Interaction of lumps in KP1
6.6.2 Solitons in the Swift-Hohenberg model
6.7 The Motion of Hydrodynamic Vortices
6.7.1 The general scheme
6.7.2 A single vortex near the interface
6.7.3 Dynamics of a vortex pair in a fluid with a density jump
References
Chapter 7 Fast and Slow Motions. Autowaves
7.1 Non-Dispersive Nonlinear Waves
7.1.1 Burgers equation and Taylor shocks
7.1.2 Periodic simple wave with discontinuities
7.1.3 Waves in media with hysteresis
7.2 Nonlinear Wave Propagation Along the Rays
7.2.1 Simple wave with variable parameters
7.2.2 Spherical nonlinear waves
7.2.3 Upward propagation in the atmosphere
7.3 Autosolitons and Explosive Instability
7.4 Interaction of Solitons with a Long Wave
7.4.1 Amplification and generation of solitons
7.4.2 Two-dimensional resonators
7.5 A Soliton on a Long Wave in a Rotating Fluid
7.5.1 General relations
7.5.2 Conservative case: no radiation
7.5.3 Effect of radiative losses
7.6 Autowaves in Reaction-Diffusion Systems
7.6.1 KPP–Fisher model
7.6.2 A two-component model
7.7 Spatial Structure of Field in a Laser Medium
References
Chapter 8 Direct Asymptotic Perturbation Theory
8.1 Perturbation Method for Quasi-Harmonic Waves
8.1.1 General scheme
8.1.2 Resonant interaction of waves
8.2 The Method for Non-Sinusoidal Periodic Waves
8.2.1 General scheme
8.3 Lagrangian Systems
8.4 Averaged Lagrangian and Whitham’s Variational Principle
8.5 Linear Waves
8.6 Perturbation Method for Lumped-Parameter Systems
8.6.1 Quasi-harmonic oscillations
8.6.2 Non-sinusoidal oscillations
8.6.3 Lagrangian ODE systems
8.7 Perturbation Method for Solitary Waves and Fronts
8.7.1 General scheme
8.7.2 Lagrangian description of solitons
8.7.3 A scheme for interacting solitons
8.7.4 Interaction of solitons in Lagrangian systems
8.7.5 Notes on the systems close to exactly integrable
References
Epilogue
Index
