Padé approximants and nonlinear waves

1979 ◽  
Vol 21 (3) ◽  
pp. 549-571 ◽  
Author(s):  
F. J. Romerias ◽  
J. P. Dougherty
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Nonlinear Waves ◽  
Wave Amplitude ◽  
Plane Waves ◽  
Electron Plasma ◽  
Cold Electron ◽  
Small Wave ◽  
The Waves ◽  
Approximant Method

The perturbation solution of the ordinary differential equations that describe exact nonlinear travelling plane waves leads to asymptotic expansions in powers of the (small) wave amplitude for both the proffle and the frequency of the waves. This paper shows how the Padé approximant method can be used to extend the validity of those expansions to larger amplitudes. The method is applied to the Duffing equation and to two types of nonlinear waves in a cold electron plasma: longitudinal oscillations and coupled transverse–longitudinal relativistic waves.

Further analysis of nonlinear, periodic, highly superluminous waves in a magnetized plasma

1982 ◽  
Vol 27 (1) ◽  
pp. 177-187 ◽  
Author(s):  
P. C. Clemmow
Keyword(s):  
Magnetic Field ◽  
Power Series ◽  
Periodic Solutions ◽  
Wave Speed ◽  
Nonlinear Differential Equations ◽  
Plane Waves ◽  
Electron Plasma ◽  
Finite Amplitude ◽  
Magnetized Plasma ◽  
Cold Electron

A perturbation method is applied to the pair of second-order, coupled, nonlinear differential equations that describe the propagation, through a cold electron plasma, of plane waves of fixed profile, with direction of propagation and electric vector perpendicular to the ambient magnetic field. The equations are expressed in terms of polar variables π, φ, and solutions are sought as power series in the small parameter n, where c/n is the wave speed. When n = 0 periodic solutions are represented in the (π,φ) plane by circles π = constant, and when n is small it is found that there are corresponding periodic solutions represented to order n2 by ellipses. It is noted that further investigation is required to relate these finite-amplitude solutions to the conventional solutions of linear theory, and to determine their behaviour in the vicinity of certain resonances that arise in the perturbation treatment.

Stability of strong electromagnetic waves in overdense plasmas

1982 ◽  
Vol 27 (2) ◽  
pp. 239-259 ◽  
Author(s):  
F. J. Romeiras
Keyword(s):  
Nonlinear Waves ◽  
Electromagnetic Waves ◽  
Longitudinal Direction ◽  
Electron Plasma ◽  
Cold Electron ◽  
Small Perturbations ◽  
Circularly Polarized ◽  
Instability Growth ◽  
Linear Differential ◽  
The Stability

We consider the stability against small perturbations of a class of exact wave solutions of the equations that describe an unmagnetized relativistic cold electron plasma. The main feature of these nonlinear waves is a transverse circularly polarized electric field with periodic amplitude modulation in the longitudinal direction. Floquet's theory of linear differential equations with periodic coefficients is used to solve the perturbation equations and obtain the instability growth rates.

Periodic, highly superluminous, nonlinear waves in a cold, unmagnetized plasma

1982 ◽  
Vol 27 (2) ◽  
pp. 267-276 ◽  
Author(s):  
P. C. Clemmow
Keyword(s):  
Perturbation Method ◽  
Nonlinear Waves ◽  
Small Amplitude ◽  
Plane Waves ◽  
Electron Plasma ◽  
Periodic Wave ◽  
Electric Vector ◽  
Third Wave ◽  
Transverse Waves ◽  
Unmagnetized Plasma

With respect to the propagation through a cold, unmagnetized, electron plasma of nonlinear, highly superluminous, plane waves of fixed profile, with electric vector in a fixed plane parallel to the direction of propagation, it is known that, in addition to the familiar longitudinal and quasi-transverse waves, there can also be a third periodic wave. The perturbation method by which this third wave has previously been analysed is of restricted validity, and fails to describe how the wave disappears in the approach to the small-amplitude limit, where the longitudinal and transverse waves alone survive.

Anomalous absorption of the circularly polarized electromagnetic beams near the plasma frequency in a magnetized electron plasma

2007 ◽  
Vol 73 (3) ◽  
pp. 315-330 ◽  
Author(s):  
S. R. SESHADRI
Keyword(s):  
Resonant Frequency ◽  
Plasma Frequency ◽  
Unit Length ◽  
Plane Waves ◽  
Electron Plasma ◽  
Magnetostatic Field ◽  
Power Absorption ◽  
Circularly Polarized ◽  
Focused Beams ◽  
Electromagnetic Beams

AbstractThe propagation of circularly polarized electromagnetic beams along the magnetostatic field in an electron plasma is investigated. As a consequence of a strong interaction with the medium, the beam spreads rapidly on propagation near the cutoff frequencies and the cyclotron resonant frequency of the corresponding plane waves, as well as near the plasma frequency. The power absorption for unit length near the cyclotron frequency and the plasma frequency are determined. For tightly focused beams, there is significant power absorption near the plasma frequency as compared with that at the cyclotron resonant frequency.

Nonlinear Corrections for Edge Bending of Shells

1980 ◽  
Vol 47 (4) ◽  
pp. 861-865 ◽  
Author(s):  
G. V. Ranjan ◽  
C. R. Steele
Keyword(s):  
Differential Equations ◽  
Spherical Shell ◽  
Asymptotic Expansions ◽  
Axial Displacement ◽  
Exponential Functions ◽  
Spherical Cap ◽  
Influence Coefficients ◽  
Edge Loading ◽  
General Geometry ◽  
Edge Influence

Asymptotic expansions for self-equilibrating edge loading are derived in terms of exponential functions, from which formulas for the stiffness and flexibility edge influence coefficients are obtained, which include the quadratic nonlinear terms. The flexibility coefficients agree with those previously obtained by Van Dyke for the pressurized spherical shell and provide the generalization to general geometry and loading. In addition, the axial displacement is obtained. The nonlinear terms in the differential equations can be identified as “prestress” and “quadratic rotation.” To assess the importance of the latter, the problem of a pressurized spherical cap with roller supported edges is considered. Results show that whether the rotation at the edge is constrained or not, the quadratic rotation terms do not have a large effect on the axial displacement. The effect will be large for problems with small membrane stresses.

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