## Nonlinear waves in waveguides: with stratification |

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Page 19

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 is ...

**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 is ...

Page 21

The equation L(ik)ij> = iwV> in the Fourier transform space has generally five

different eigenvectors and eigenvalues that depend on fe [see e.g. Ref. 2.4].

Moreover, five

P(<*>, ...

The equation L(ik)ij> = iwV> in the Fourier transform space has generally five

different eigenvectors and eigenvalues that depend on fe [see e.g. Ref. 2.4].

Moreover, five

**projection operators**P(o) can be constructed for them: (P(<*>)2 =P(<*>, ...

Page 23

We build up a general form for the

with definite values ai and bi. By calculation we state that if / t*i Pi 7. \ Pw = I aw

atfi am , (2.41) \ biai bij3i bm ) then P(i)(A, B, C)T = ^i(l,aj, 6,)T; /i = aiA+ftP+^C.

We build up a general form for the

**projection operators**to the subspaces of (2.40)with definite values ai and bi. By calculation we state that if / t*i Pi 7. \ Pw = I aw

atfi am , (2.41) \ biai bij3i bm ) then P(i)(A, B, C)T = ^i(l,aj, 6,)T; /i = aiA+ftP+^C.

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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 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 S.BXeble 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