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On the initial-value problem in a cold plasma model

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Abstract

We consider a homogeneous, weakly ionized gas filling all of space and in it the propagation of a plane electromagnetic wave induced at positive times by the electromagnetic field story at negative times and by an assigned external current. The propagation can be described by an integro-differential equation if one takes into account the collisions between molecules and electrons by a suitable linear inheritance relation. We determine explicitly the fundamental solution and therefore solve the associated classical initial value problem with completely arbitrary initial data.

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Notes

  1. In absence of collisions (\( a\!=\!0\)) the formula amounts to saying that the electron velocity is the integral of its acceleration over the previous time, and the acceleration is proportional to the electric field. In the presence of collisions such proportionality holds only between two subsequent collisions. The rationale behind (5) is that therefore the effect of the electric field at earlier and earlier times is more and more damped by the time-dependent exponential in the integral.

  2. In fact, the change of independent variable \(x\mapsto x' = \sqrt{\varepsilon \mu } \; x\) reduces \(\varepsilon \mu \) to 1; and under the above assumptions the solution for general polarization is the superposition of two linearly polarized electromagnetic fields, that are orthogonal to each other.

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Acknowledgments

This research was partially supported by UniNA and Compagnia di San Paolo under the grant “STAR Program 2013”.

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  1. Dip. di Matematica e Applicazioni, Università “Federico II”, Via Claudio 21, 80125, Naples, Italy

    G. Fiore, A. Maio & P. Renno

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  1. G. Fiore
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  2. A. Maio
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  3. P. Renno
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Correspondence to G. Fiore.

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Communicated by Salvatore Rionero.

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Fiore, G., Maio, A. & Renno, P. On the initial-value problem in a cold plasma model. Ricerche mat. 63 (Suppl 1), 157–164 (2014). https://doi.org/10.1007/s11587-014-0206-8

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