SLOWLY VARYING OSCILLATIONS AND WAVES

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.Contents:

• Preface
• Introduction
• Perturbed Oscillations
• Linear Waves
• Nonlinear Quasi-Harmonic Waves
• Modulated Non-Sinusoidal Waves
• Slowly Varying Solitons
• Interaction of Solitons, Kinks, and Vortices
• Fast and Slow Motions. Autowaves
• Direct Asymptotic Perturbation Theory
• Epilogue
• Index

Readership: Advanced undergraduate and graduate students and researchers in the fields of applied mathematics, fluid dynamics, biophysics, and others.Oscillators and Waves;Slowly Varying Oscillators and Waves;Asymptotic Perturbation Methods;Linear and Nonlinear Oscillations and Waves;Resonance;Van Der Pol Oscillator;Nonlinear Pendulum;Models of Laser;Synchronization of Oscillators;Duffing Oscillator;Kuramoto Model;Elliptic Functions;Parametric Resonance;Dispersion Relation;Group and Phase Velocities;Klein-Gordon Equation;Schrodinger Equation, Linear and Nonlinear;Self-Similar Solutions;Wave Asymptotic;Stationary Phase Method;Airy Function;Water Waves;Geometrical Theory of Waves;Wave Trapping;Waves of Envelopes;Wave Beams;Modulation Instability;Self-Focusing;Gross-Pitaevskii Equation;Frequency Doubling;Period Doubling;Nonlinear Optics;Nonlinear Acoustics;Cnoidal Waves;Korteweg-de Vries Equation;Rotational Kdv Equation;Soliton;Terminal Damping;Geometrical Theory;Average Lagrangian;Whitham's Theory;Kadomtsev-Petviashvili Equation;Kinks;Ensembles of Solitons and Kinks;Compound Solitons;Sine-Gordon Equation;Gardner Equation;Internal Waves;Swift-Hohenberg Equation;Burgers Equation;Taylor Shock;Autowaves;Autosolitons;Reaction-Diffusion System;KPP-fisher Model;Coupled Waves0Key Features:

• The book is unique in that it covers a broad range of problems related to linear and nonlinear oscillators and waves, from simplified approximations to various applications to more consistent mathematical schemes. It covers quite different problems, from classical to recently studied. Most of the contents is written at an introductory level. It may be useful for both the experts and the young students involved in physics, fluid mechanics, biophysics, and applied mathematics
• Systematic. One book allows to study a broad spectrum of problems related to oscillators and waves