Nonlinear Evolution of External Ideal MHD Modes Near the Boundary of Marginal Stability

1984 ◽  
Vol 39 (3) ◽  
pp. 288-308
Author(s):  
E. Rebhan
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Evolution ◽  
Nonlinear Boundary Conditions ◽  
Higher Order ◽  
Reductive Perturbation Method ◽  
Marginal Stability ◽  
Amplitude Equations ◽  
Full Generality ◽  
Ideal Mhd ◽  
Higher Order Corrections

AbstractThe nonlinear evolution of external ideal MHD-modes is determined from the equations of ideal MHD by employing a reductive perturbation method which uses a driving parameter for expansion. The reduction of the plasma equations is the same as for internal modes and was treated previously [1]. A main problem arising in addition for external modes is the reduction of the nonlinear boundary conditions. The set of reduced boundary conditions is obtained on the undisplaced boundary in the marginally stable equilibrium position. Another additional problem arises from the fact that the linear MHD operator is only selfadjoint for linear eigenmodes but not for the higher order mode corrections. This complicates the determination of nonlinear amplitude equations for the marginal mode which are obtained from solubility conditions. The amplitude equations are qualitatively the same as for internal modes. Quantitatively, the calculation of the coefficients in these is different. Explicit expressions for the coefficients are derived in full generality. The effect of higher order corrections to the nonlinear amplitude equations is discussed quantitatively for one of two possible cases and qualitatively for the other.

Effect of Higher-Order Corrections on the Propagation of Nonlinear Dust-Acoustic Solitary Waves in Mesospheric Dusty Plasmas

2006 ◽  
Vol 61 (7-8) ◽  
pp. 316-322 ◽  
Author(s):  
Sayed A. Elwakil ◽  
Mohamed T. Attia ◽  
Mohsen A. Zahran ◽  
Emad K. El-Shewy ◽  
Hesham G. Abdelwahed
Keyword(s):  
Acoustic Waves ◽  
Kdv Equation ◽  
Type Equation ◽  
Dusty Plasmas ◽  
Higher Order ◽  
Reductive Perturbation Method ◽  
Dust Acoustic Solitary Waves ◽  
Dust Acoustic ◽  
Renormalization Method ◽  
Higher Order Corrections

The contribution of the higher-order correction to nonlinear dust-acoustic waves are studied using the reductive perturbation method in an unmagnetized collisionless mesospheric dusty plasma. A Korteweg - de Vries (KdV) equation that contains the lowest-order nonlinearity and dispersion is derived from the lowest order of perturbation, and a linear inhomogeneous (KdV-type) equation that accounts for the higher-order nonlinearity and dispersion is obtained. A stationary solution is achived via renormalization method

The Effect of Higher-Order Corrections on the Propagation of Nonlinear Dust-Acoustic Solitary Waves in a Dusty Plasma with Nonthermal Ions Distribution

2008 ◽  
Vol 63 (5-6) ◽  
pp. 261-272 ◽  
Author(s):  
Hesham G. Abdelwahed ◽  
Emad K. El-Shewy ◽  
Mohsen A. Zahran ◽  
Mohamed T. Attia
Keyword(s):  
Dusty Plasma ◽  
Kdv Equation ◽  
Higher Order ◽  
Reductive Perturbation Method ◽  
Charged Dust ◽  
Ion Distributions ◽  
Kdv Equations ◽  
Dust Acoustic ◽  
Renormalization Method ◽  
Higher Order Corrections

Propagation of nonlinear dust-acoustic (DA) waves in a unmagnetized collisionless mesospheric dusty plasma containing positively and negatively charged dust grains and nonthermal ion distributions are investigated. For nonlinear DA waves, a reductive perturbation method is employed to obtain a Korteweg-de Vries (KdV) equation for the first-order potential. As it is well-known, KdV equations contain the lowest-order nonlinearity and dispersion, and consequently can be adopted for only small amplitudes. As the wave amplitude increases, the width and velocity of a soliton can not be described within the framework of KdV equations. So, we extend our analysis and take higher-order nonlinear and dispersion terms into account to clarify the essential effects of higher-order corrections. Moreover, in order to study the effects of higher-order nonlinearity and dispersion on the output solution, we address an appropriate technique, namely the renormalization method.

Higher order nonlinear and dispersive effects on ion-acoustic solitary waves

1991 ◽  
Vol 69 (7) ◽  
pp. 822-827 ◽  
Author(s):  
K. P. Das ◽  
S. R. Majumdar
Keyword(s):  
Solitary Wave ◽  
Higher Order ◽  
Reductive Perturbation Method ◽  
Inhomogeneous Equation ◽  
Governing Equations ◽  
First Order ◽  
Ion Acoustic Solitary Waves ◽  
Reductive Perturbation ◽  
Ion Acoustic ◽  
Higher Order Corrections

Starting from an integrated form of the system of governing equations in terms of pseudopotential, higher order nonlinear and dispersive effects are obtained for an ion-acoustic solitary wave. The advantage of the method developed here is that instead of solving a second-order inhomogeneous differential equation at each order in the reductive perturbation method, we are to solve a first-order inhomogeneous equation at each order. Expressions are obtained for both the Mach number and the width of the solitary wave as functions of amplitude, including higher order corrections.

Ordinary Differential Equations with Nonlinear Boundary Conditions

2002 ◽  
Vol 9 (2) ◽  
pp. 287-294
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Lower And Upper Solutions ◽  
Iterative Technique ◽  
Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

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