The propagation of weakly nonlinear waves in a plasma including low-pressure cosmic rays

1990 ◽  
Vol 164 (2) ◽  
pp. 183-191 ◽  
Author(s):  
S. V. Chalov
Keyword(s):  
Cosmic Rays ◽  
Nonlinear Waves ◽  
Low Pressure ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Waves

scholarly journals Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2. Stability of cnoidal waves

2007 ◽  
Vol 73 (6) ◽  
pp. 933-946
Author(s):  
S. PHIBANCHON ◽  
M. A. ALLEN ◽  
G. ROWLANDS
Keyword(s):  
Growth Rate ◽  
Nonlinear Waves ◽  
Electron Distribution ◽  
Magnetized Plasma ◽  
Weakly Nonlinear ◽  
Long Wavelength ◽  
First Order ◽  
Wave Solutions ◽  
Linear Waves ◽  
Weakly Nonlinear Waves

AbstractWe determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the Schamel–Korteweg–de Vries–Zakharov–Kuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to a minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero, and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio b increases as the electron distribution becomes increasingly flat-topped. As b and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle θ at which the perturbation is applied. Solutions whose minimum values are zero and which travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of θ for which the first-order growth rate is not zero.

PROPAGATION OF WEAKLY NONLINEAR WAVES IN A FLUID-FILLED THIN ELASTIC TUBE

1999 ◽  
Vol 23 (2) ◽  
pp. 253-265
Author(s):  
H. Demiray
Keyword(s):  
Nonlinear Waves ◽  
Vries Equation ◽  
Perturbation Technique ◽  
Fluid Pressure ◽  
Static Field ◽  
Elastic Tube ◽  
Inviscid Fluid ◽  
Weakly Nonlinear ◽  
De Vries ◽  
Weakly Nonlinear Waves

In this work, we study the propagation of weakly nonlinear waves in a prestressed thin elastic tube filled with an inviscid fluid. In the analysis, considering the physiological conditions under which the arteries function, the tube is assumed to be subjected to a uniform pressure P0 and a constant axial stretch ratio λz. In the course of blood flow in arteries, it is assumed that a finite time dependent radial displacement is superimposed on this static field but, due to axial tethering, the effect of axial displacement is neglected. The governing nonlinear equation for the radial motion of the tube under the effect of fluid pressure is obtained. Using the exact nonlinear equations of an incompressible inviscid fluid and the reductive perturbation technique, the propagation of weakly nonlinear waves in a fluid-filled thin elastic tube is investigated in the longwave approximation. The governing equation for this special case is obtained as the Korteweg-de-Vries equation. It is shown that, contrary to the result of previous works on the same subject, in the present work, even for Mooney-Rivlin material, it is possible to obtain the nonlinear Korteweg-de-Vries equation.

scholarly journals Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 1. Stability of solitary waves

2007 ◽  
Vol 73 (2) ◽  
pp. 215-229 ◽  
Author(s):  
M.A. ALLEN ◽  
S. PHIBANCHON ◽  
G. ROWLANDS
Keyword(s):  
Growth Rate ◽  
Solitary Waves ◽  
Nonlinear Waves ◽  
Electron Distribution ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Magnetized Plasma ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Waves ◽  
Stability Of Solitary Waves

Abstract.Weakly nonlinear waves in strongly magnetized plasma with slightly non-isothermal electrons are governed by a modified Zakharov–Kuznetsov (ZK) equation, containing both quadratic and half-order nonlinear terms, which we refer to as the Schamel–Korteweg–de Vries–Zakharov–Kuznetsov (SKdVZK) equation. We present a method to obtain an approximation for the growth rate, γ, of sinusoidal perpendicular perturbations of wavenumber, k, to SKdVZK solitary waves over the entire range of instability. Unlike for (modified) ZK equations with one nonlinear term, in this method there is no analytical expression for kc, the cut-off wavenumber (at which the growth rate is zero) or its corresponding eigenfunction. We therefore obtain approximate expressions for these using an expansion parameter, a, related to the ratio of the nonlinear terms. The expressions are then used to find γ for k near kc as a function of a. The approximant derived from combining these analytical results with the ones for small k agrees very well with the values of γ obtained numerically. It is found that both kc and the maximum growth rate decrease as the electron distribution becomes progressively less peaked than the Maxwellian. We also present new algebraic and rarefactive solitary wave solutions to the equation.

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