Phibanchon, S., Allen, M. A. and Rowlands, G. (George). (2007) Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2, Stability of cnoidal waves. Journal of Plasma Physics, Vol.73 (No.6). pp. 933-946. ISSN 0022-3778

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Abstract

We 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.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Library of Congress Subject Headings (LCSH):	Maxwell equations -- Numerical solutions, Electromagnetic theory -- Mathematics, Nonlinear waves -- Mathematics, Nonlinear theories
Journal or Publication Title:	Journal of Plasma Physics
Publisher:	Cambridge University Press
ISSN:	0022-3778
Date:	21 February 2007
Volume:	Vol.73
Number:	No.6
Page Range:	pp. 933-946
Identification Number:	10.1017/S002237780700640X
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	Allen, M. A., Phibanchon, S. and Rowlands, G. 2006 J. Plasma Phys. Published online 23 May 2006, doi:10.1017/S0022377806004508. Byrd, P. F. and Friedman, M. D. 1954 Handbook of Elliptic Integrals for Engineers and Physicists. Berlin: Springer. Infeld, E. 1985 J. Plasma Phys. 33, 171. Infeld, E. and Rowlands, G. 2000 Nonlinear Waves, Solitons and Chaos, 2nd edn. Cambridge: Cambridge University Press. Munro, S. and Parkes, E. J. 1999 J. Plasma Phys. 62, 305. O'Keir, I. S. 1993 The stability of solutions to modified generalized Korteweg–de Vries, nonlinear Schrödinger and Kadomtsev–Petviashvili equations. PhD thesis, University of Strathclyde. O'Keir, I. S. and Parkes, E. J. 1997 Phys. Scripta 55, 135. Parkes, E. J. 1993 J. Phys. A: Math. Gen. 26, 6469. Phibanchon, S. 2006 Nonlinear waves in plasmas with trapped electrons. PhD thesis, Mahidol University. Rowlands, G. 1969 J. Plasma Phys. 3, 567. Schamel, H. 1972 Plasma Phys. 14, 905. Schamel, H. 1973 J. Plasma Phys. 9, 377. Zemanian, A. H. 1965 Distribution Theory and Transform Analysis. New York: McGraw-Hill.
URI:	http://wrap.warwick.ac.uk/id/eprint/663

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