Abstract
We apply the nonlinear Fourier analysis developed by Callebaut to an infinite homogeneous plasma calculating many higher order terms (computer algebra) and obtaining in this way some analytic expressions. (a) For cold plasma: the maximum amplitude is 2/e (i.e., 73% of n0) of the initial density n0, otherwise the series diverges. For exponentials (sum of two waves) the maximum amplitude is halved, i.e. n0/e. (b) For plasma with electron pressure, the radius of convergence decreases as the ratio of k2vs-2(1 + Γ-)/ω-2 increases (Γ- is the polytropic exponent; ω- is the plasma angular frequency for electrons; k is the wave number; vs- is the sound velocity for the electrons). (c) Suggestions for experimental verification are made. (d) In the limit of sound waves (no plasma) the radius of convergence is zero. Nevertheless the correct dispersion relation is obtained. A direct analysis confirmed these results for sound waves. (e) The cases where the method fails are indicated. (f) Plasma where both ions and electrons may move, are briefly considered (relevant for comet tails, fullerenes and electron–positron plasmas).
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