## Nonlinear Waves in Waveguides: with StratificationS.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. |

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Page 19

2.2

Type For the sake of mathematical simplicity it is useful to adopt the idea of a

dispersion relation branch - the subspace of solutions in which the time

dependence ...

2.2

**Projection Operators**on Subspaces of Waves 2.2.1 Determination of WaveType For the sake of mathematical simplicity it is useful to adopt the idea of a

dispersion relation branch - the subspace of solutions in which the time

dependence ...

Page 21

2.4]. Moreover, five

= P<°>, p<«>pO»> = 0, £a P(a) = J. Examples of such operators are discussed

above. The explicit forms for the eigenvectors and operators P(a) in a general ...

2.4]. Moreover, five

**projection operators**PM can be constructed for them: (p(">)2= P<°>, p<«>pO»> = 0, £a P(a) = J. Examples of such operators are discussed

above. The explicit forms for the eigenvectors and operators P(a) in a general ...

Page 23

Y?i=\ $i = X)Li P(,)^- We build up a general form for the

subspaces of (2.40) with definite values a; and By calculation we state that if / «i

Pi 7. \ P(,) = I a,a; aiPi ani J , (2.41) \ bicti bi0i bm ) then pW(A, B, C)T = ^(1,^, bi)T

...

Y?i=\ $i = X)Li P(,)^- We build up a general form for the

**projection operators**to thesubspaces of (2.40) with definite values a; and By calculation we state that if / «i

Pi 7. \ P(,) = I a,a; aiPi ani J , (2.41) \ bicti bi0i bm ) then pW(A, B, C)T = ^(1,^, bi)T

...

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### Contents

Introduction | 1 |

The Discrimination and Interaction | 12 |

Interaction of Modes in an Electromagnetic Field Waveguide | 50 |

Copyright | |

6 other sections not shown

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### Common terms and phrases

allows amplitude approximation atmosphere atmospheric waveguide atmospheric waves basis functions boundary conditions calculation CKdV coefficients components considered contribution coordinate decrease denote density density matrix dependence derivation described determined dielectric dimensionless dispersion branches dispersion equation dispersion relation dissipation distribution function dynamical variables effects evolution equations exponential Fiz.Atm.Okean formulas Fourier frequency given hydrodynamical inhomogeneity initial conditions integration internal waves ion-acoustic ionospheric iteration Kaliningrad KdV equation kinetic Langmuir waves layer linear long waves magnetic field matrix mean field medium method mode interaction mode number Moscow nonlinear constants nonlinear terms Nonlinear Waves nonlocal ocean oscillations perturbation theory physical plasma waves problem projection operators quasi-waveguide quasisolitons region resonance Rossby waves S.B.Leble scale Sect small parameters soliton solution spectral SSSR stationary stratified subspaces substitution taking into account temperature thermoclyne thermoconductivity thermospheric three-wave transformed turbulence two-dimensional values velocity vertical wave interaction wave propagation wave vector waveguide propagation wavelength