can obtain a higher order differential equation that involves only the electric field,

$$a\partial _{x}^{4}E+d\partial _{x}^{2}E+cE=0,  \eqnum{3.16}$$

Here, $a=\omega \kappa $, $d=\omega \eta +\alpha \kappa -\gamma -\tau \kappa
\chi $, and $c=\alpha \eta -\sigma \gamma -\tau \eta \chi $.

Solving (3.16) we find

$$E\left( x\right) =C_{1}e^{-x\xi _{1}}+C_{2}e^{x\xi _{1}}+C_{3}e^{-x\xi
_{2}}+C_{4}e^{x\xi _{2}},  \eqnum{3.17}$$

where

$$\xi _{1,2}=\left[ -d\pm \sqrt{d^{2}-4ac}\right] ^{\frac{1}{2}}/\sqrt{2a} 
\eqnum{3.18}$$

When the low density approximation $n_{e}\ll m^{2}$ is considered (this approximation is in agreement with the typical values $n_{e}=2\times
10^{14}cm^{-2}$, $m_{e}=2.6\times 10^{10}cm^{-1}$ found in high-T$_{C}$ superconductivity), we find that, for $N=2$ and at temperatures lower than the energy gap $\left( T\ll \omega _{c}\right) $, the inverse length scales (3.18) are given by the following real functions $$\xi _{1}\simeq e\sqrt{\frac{m}{\pi }}\left[ 1+\left( \frac{\pi ^{2}n_{e}^{2}%
}{m^{3}}\right) \beta \exp -\left( \frac{\pi n_{e}\beta }{2m}\right) \right]
\eqnum{3.19}$$ $$\xi _{2}\simeq e\sqrt{\frac{n_{e}}{m}}\left[ 1-\left( \frac{\pi n_{e}}{m}%
\right) \beta \exp -\left( \frac{\pi n_{e}\beta }{2m}\right) \right] 
\eqnum{3.20}$$ These two inverse length scales correspond to two short-range electromagnetic modes of propagation. These results are consistent with those obtained in ref. [@8] using a different approach. If the masses of the two massive modes, obtained in ref. [@8] by using the electromagnetic thermal Green’s function for static and slowly varying configurations, are evaluated in 