Nonlinear Waves in Waveguides

S.B. Leble's book deals with nonlinear waves and their propagation in metallic and dielectric waveguides and media with stratification. The underlying nonlinear evolution equations (NEEs) are derived giving also their solutions for specific situations. The reader will find new elements to the traditional approach. Various dispersion and relaxation laws for different guides are considered as well as the explicit form of projection operators, NEEs, quasi-solitons and of Darboux transforms. Special points relate to: 1. the development of a universal asymptotic method of deriving NEEs for guide propagation; 2. applications to the cases of stratified liquids, gases, solids and plasmas with various nonlinearities and dispersion laws; 3. connections between the basic problem and soliton- like solutions of the corresponding NEEs; 4. discussion of details of simple solutions in higher- order nonsingular perturbation theory.

Contenu

1. Introduction.- 1.1 General Remarks.- 1.2 Overview.- 2. The Discrimination and Interaction of Hydrodynamical Waves of Different Types.- 2.1 Hydrodynamic Wave Equations in a Three-Dimensional Stratified Medium.- 2.1.1 Fundamentals.- 2.1.2 Atmospheric Waves: Dispersion and Polarization.- 2.2 Projection Operators on Subspaces of Waves.- 2.2.1 Determination of Wave Type.- 2.2.2 Projection Operators in Geophysical Hydrodynamics.- 2.2.3 Projection Operators for Poincare and Rossby Waves.- 2.3 Interaction of Surface Poincare and Rossby Waves.- 2.4 Long Atmospheric Internal Waves.- 2.4.1 Internal Waves in an Atmospheric Waveguide.- 2.5 The Method of Successive Approximations.Solution for the Cauchy Problem of the Coupled Korteweg-de Vries Equations.- 2.6 The Coupled KdV Equations.- 2.6.1 Quasisolitons of CKdV.- 2.6.2 The Symmetry and Integrability of CKdV.- 2.6.3 The Stationary Solutions of CKdV.- 2.7 Nonlinear Rossby Waves on a Sphere.- 3. Interaction of Modes in an Electromagnetic Field Waveguide.- 3.1 Electromagnetic Fields in Dielectric Waveguides.- 3.1.1 Basic Equations and Boundary Conditions.- 3.1.2 Waveguide Modes.- 3.2 Equations of Mode Interaction.- 3.3 Three-Wave Interaction of Magnetic Modes in an Extended System of Nuclear Spins.- 3.4 Nonlinear Mode Dispersion in Dielectric Waveguides.- 4. Nonlinear Waves in Stratified Plasma.- 4.1 Wave Modes in Stratified Plasma.- 4.1.1 Description of Plasma Waves.- 4.1.2 Weakly Inhomogeneous Bounded Plasma.- 4.2 Interaction of Plasma Waves.- 4.2.1 Langmuir and Ion Wave Interactions.- 4.2.2 Three-Wave Helico-Acoustic Interaction.- 4.3 Statistical Averaging and Weak Turbulence in Plasma.- 4.3.1 Spectral Densities of Electromagnetic Field Fluctuations.- 4.3.2 One-Dimensional Turbulence.- 4.4 Multisoliton Solutions of Discrete Silin-Tikhonchuk Equations.- 4.4.1 The Darboux-Matveev Transformation.- 5. Evolution Equations for Internal Waves in Media with Strongly Inhomogeneous Stratification.- 5.1 Waves in a Medium with Varying Stratification.- 5.2 Analogues of the KdV and KP Equations with Nonlocal Dispersion.- 5.3 Quasi-Waveguide Propagation of Internal Waves in the Atmosphere.- 5.4 Waves in Gases Inhomogeneous in Knudsen Number.- 6. Mean Field Generation by Waves in a Dissipative Medium.- 6.1 Description of a Wave-Medium System by the Scales of Motion.- 6.2 Thermoactivity and Diffusion in the Presence of Waves.- 6.3 Reaction of Ionosphere Plasma to a Passing Internal Wave ....- 6.4 Interaction of Internal Waves with a Dissipative Medium.- 6.4.1 Temperature Field Instability Induced by an Internal Wave.- 6.4.2 Nonsingular Perturbation Theory.- 6.5 Mean Field Generation in a Dissipative Medium and Fine Structure of the Oceanic Thermoclyne.- Appendix 1 Atmospheric Waves over a Rotating Planet.- Appendix 2 Nonlinear Terms for Interacting Modes of Poincare and Rossby Waves at a Rotating Channel Surface.- Appendix 3 Projection to the Eigen Subspaces for Acoustic and Internal Waves.- Appendix 4 Basis Vectors, Interaction Operator for Atomic Nuclei with Spin 1.- Appendix 5 Nonlinear Internal Waves and Shear Flow Interaction Constants.- References.