Non-Linear Waves in Dispersive Media

International Series of Monographs in Natural Philosophy

1st Edition - January 1, 1975
Author: V. I. Karpman
Editor: D. ter Haar
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 8 7 1 5 - 0

Non-Linear Waves in Dispersive Media introduces the theory behind such topic as the gravitational waves on water surfaces. Some limiting cases of the theory, wherein proof of an… Read more

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Non-Linear Waves in Dispersive Media introduces the theory behind such topic as the gravitational waves on water surfaces. Some limiting cases of the theory, wherein proof of an asymptotic class is necessary and generated, are also provided. The first section of the book discusses the notion of linear approximation. This discussion is followed by some samples of dispersive media. Examples of stationary waves are also examined. The book proceeds with a discussion of waves of envelopes. The concept behind this subject is from the application of the methods of geometrical optics to non-linear theory. A section on non-linear waves with slowly varying parameters is given at the end of the book, along with a discussion of the evolution of electro-acoustic waves in plasma with negative dielectric permittivity. The gravitational waves on fluid surfaces are presented completely. The text will provide valuable information for physicists, mechanical engineers, students, and researchers in the field of optics, acoustics, and hydrodynamics.

Preface

Introduction

1 Linear Approximation

§ 2 General Solution of the Linearized Equations

§ 3 Linearized Korteweg-de Vries Equation

2 Examples of Dispersive media

§ 4 Gravitational Waves on Fluid Surfaces

§ 5 The Boussinesq Equation

§ 6 Ion-Sound Waves in Unmagnetized Plasma

§ 7 Non-linear Waves in Magnetized Plasma

§ 8 Non-linear Electromagnetic Waves in Isotropic Dielectrics

§ 9 Sound Waves with Dispersion

3 Non-linear Stationary Waves

§ 10 Steady Solutions of the Boussinesq Equations

§ 11 Stationary Waves Propagating Transversely to the Magnetic Field in Rarefied Plasma

§ 12 Other Examples of Stationary Waves

4 Non-linear Waves in Weakly Dispersive Media

§ 13 The Burgers Equation

§ 14 Solution of the Burgers Equation

§ 15 The Korteweg-de Vries Equation

§ 16 Conservation Laws for the Korteweg-de Vries Equation

§ 17 General Pattern of the Evolution of Initial Perturbations in Weakly Dispersive Media

§ 18 Analytical Solution of the Korteweg-de Vries Equation

§ 19 Asymptotic Expressions for the Amplitudes of Solitons and "Tails" for Large Values of δ

§ 20 Self-Similar Solutions of the Korteweg-de Vries Equation

§ 21 Quasi-Linear Solutions of the Korteweg-de Vries Equation

§ 22 Flow Around a Thin Body in a Dispersive Medium

§ 23 Shock Waves in Dispersive Media

5 Waves of Envelopes

§ 24 Non-linear Geometrical Optics

§ 25 Instability Criteria for Stationary Waves

§ 26 Evolution of the Wave Envelopes in the Hydrodynamic Approximation

§ 27 Non-linear Parabolic Equation

§ 28 Self-Modulation of Waves (Modulational Instability)

§ 29 Self-Focusing and Self-Channeling of Waves

§ 30 Electro-Acoustic Waves in Plasma

Appendix A Non-linear Waves with Slowly Varying Parameters (Adiabatic Approximation of Whitham)

A1 Variation Principle

A2 Adiabatic Invariants

A3 Non-linear Geometrical Optics

Appendix B Evolution of Electro-Acoustic Waves in Plasma with Negative Dielectric Permittivity

B1 Boundary Conditions

B2 Excitation and Evolution of Electro-Acoustic Waves

B3 Solution of the Boundary-Value Problem

B4 General Solution of the Fundamental Equations

References

Index

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