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Nonlinear electromagnetic waves in electron-positron plasmas

N. F. Cramer
School of Physics, The University of Sydney, N.S.W. 2006, Australia
Abstract. Plasmas with unequal numbers of electrons and positrons are of interest for pulsar magnetospheres and laboratory
pair plasma experiments. Here the dispersion relation for electromagnetic waves in such a plasma is discussed, and the types
of nonlinear waves that arise on the various branches of the dispersion curves. An excess of one type of charge carrier
leads to circular and elliptical polarization, rather than linear polarization at frequencies well below the gyrofrequency, with
a corresponding fundamental difference in the type of nonlinear wave equations describing waves of moderate amplitude.
Known equations such as the derivative nonlinear Schrödinger equation are applicable to some modes of the dispersion curve,
while modified equations apply to other modes. Using a nonlinear Schrödinger equation model, the dispersive and nonlinear
coefficients are calculated, and criteria for modulational instability of nonlinear plane wave solutions are derived.
Keywords: plasma,electrons,positrons,nonlinear,electromagnetic
PACS: 52.27.Ep,52.35.Mw,97.60.Gb
Neutron stars are thought to be surrounded by a relativistic plasma of electrons and positrons, which is penetrated by
beams of high energy electrons and positrons traveling along the magnetic field. The pulsar emission mechanism may
depend on coherent nonlinear waves arising in the plasma. Previous work has concentrated on nonlinear electrostatic
modes [1], but here we analyse possible nonlinear electromagnetic modes arising in the plasma. Imbalanced electron
and positron number densities occur in the rotating pulsar magnetosphere, due to the Goldreich-Julian charge, and we
allow for unequal electron and positron number densities here. This leads to circularly polarized, rather than linearly
polarized, modes propagating along the magnetic field. The linear dispersion relation is discussed, then nonlinear
effects are analysed, and the modulational instability of nonlinear plane waves is discussed.

Weakly nonlinear waves behave generically determined by the local dispersion relation. The great variation in the
dispersion relation shown by the waves in the pair plasma indicate a variety of nonlinear wave behaviour, according to
the linear wave dispersion in a particular frequency range of interest. For example, in the range shown in Figure 1(b),
dispersion due to η = 0 modifies the Alfvén wave dispersion relation. In a normal plasma, dispersion due to the Hall
or cyclotron effect produces a derivative Nonlinear Schrödinger equation (DNLS) describing a weakly nonlinear wave
[2]. Here however the dispersive effect is different: for example the initial parabolic dependence on k z is indicative of
a whistler-type wave.
A weakly nonlinear wave of long wavelength can generally be described by a nonlinear Schrödinger equation
(NLS) [3]. Such an equation has been derived for a general multi-species plasma by Irie and Ohsawa (2001) [4], and
the resulting dispersive.

The dispersive and nonlinear coefficients for the η =0 cases differ both quantitatively and qualitatively from the η =0
(linearly polarized) case. The scale size and speed of soliton solutions are determined by α and β . We see from Fig 2
that for η = 0 a modulational instability of the plane wave solution occurs (αβ > 0) for f <∼
0.95, but that the wave is
stable for f greater than the upper cutoff, until f approaches 2. When there is an imbalance in the electron and positron
number densities (η >0), the instability criterion is largely unchanged near f =1, but at low frequencies both the LHP
and RHP modes become stable (where β becomes negative. The left-hand polarized wave has a cutoff frequency at
k = 0, and is stable for a small range of frequencies above the cutoff. The right-hand polarized wave has no cutoff at
k = 0, and remains stable for a small range of frequencies above zero. If the electron number density is greater than
the positron number density (η < 0), the polarizations of the modes discussed above become reversed.
1. D. B. Melrose, and M. E. Gedalin, Astrophys. J., 521,351–361 (1999).
2. N. F. Cramer, The Physics of Alfvén Waves, Wiley-VCH, Berlin, 2001, pp. 144–149.
3. A. Hasegawa, Phys. Fluids, 15, 870–881 (1972).
4. S. Irie, and Y. Ohsawa, J. Phys. Soc. Japan, 70, 1585–1592 (2001).
5. E. Mjølhus, and J. Wyller, Physica Scripta, 33, 442 (1986).
6. T. Taniuti, and N. Yajima, J. Math. Phys., 10, 1369 (1969).

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