## 5 Damping of Plasmon Oscillations in Ionic Systems
#### Exact Inclusion of the Lorentz Damping to the Attenuation of Ionic Dipole Surface Plasmons
Let us emphasize that the equation (36) has in general two types of particular solutions, \(e^{i\Omega t^{\prime}}\), with complex self-frequencies \(\Omega\). The solutions given by \(\Omega_{2}\) and \(\Omega_{3}\) are of oscillating type with damping (\(i\Omega_{2}\) and \(i\Omega_{3}\) are mutually conjugated, thus \(\Omega_{2}\) and \(\Omega_{3}\) have the real parts of opposite sign, whereas the same imaginary parts, the latter is positive displaying the damping rate) and the second one--given by \(\Omega_{1}\), which turns out to be an unstable exponentially rising solution (negative imaginary solution). This unstable solution is the well-known artifact in the Maxwell electrodynamics (cf. e.g., chapter 75 in [8]) and corresponds to the infinite self-acceleration of the free charge due to Lorentz friction force. More specifically, this artificial instability illustrates the singular solution of the equation \(m\dot{\mathbf{v}}=const.\times\dot{\mathbf{v}}\), which is associated with a formal renormalization of the field-mass of the charge--infinite for point-like charge and canceled in an artificial manner by arbitrary assumed negative infinite non-field mass, resulting in ordinary mass of e.g., an electron, which is, however, not defined mathematically in a proper way. This unphysical singular particular solution should be thus discarded. The other oscillatory type solution resembles the solution of the ordinary damped harmonic oscillator, though with distinct attenuation rate and frequency. They are expressed by analytical formulae for \(\Omega_{2}\) and \(\Omega_{3}\) by Equations (37) and then are calculated for various \(a\) and compared with the corresponding quantities found within the perturbation approach. This comparison is presented in Figure 4. From this comparison it is clearly visible that application of the perturbation approach leads to high overestimation of the damping rate for \(a>\bar{a}\). Therefore, we can conclude that the usage of the approximate formula for the Lorentz friction damping in the form (35) is justified up to \(a\simeq\bar{a}\), while for \(a>\bar{a}\), these approximate values strongly differ from the exact ones. The value \(\bar{a}<a_{limit}\) sharply depends on ionic system parameters and approximately \(\bar{a}\simeq\frac{a_{limit}}{2}\). Figure 4: Comparison of the damping rate and the resonance frequency (transformed also in the resonance photon wavelength—in right panels), i.e., the damping rate and frequency (wavelength) of oscillating solution of Equation (36), exact (red line) and approximate upon perturbation approach (blue line), both with respect to the ionic finite system radius \(a\). Plasmon-Polariton in the Chain of Finite Ionic Systems--Model of the Saltatory Conduction in Myelinated Axons  
The effective and quick transduction of the action potential through long myelinated axons in neurons of the peripheral nervous system and in the white matter of the central nervous system undeniably stands behind the motorics and other communication functions of the body. The acceleration of the signal transduction is associated here with jumping of it between consecutive so-called Ranvier nodes separated by myelinated sectors of the length of ca. 100 \(\upmu\)m. The nature of this signal jumping, called as the saltatory conduction, is not explained as of yet as its timing exceeds conventional models of electrical signaling in neurons. Despite the very deep current recognition in mechanisms of the nerve functioning, including the action potential spike formation mechanism on the Ranvier node [9] and the model of the diffusion-like conduction of ion current in dendrites and non-myelinated axons according to the so-called cable theory originated by [PERSON] (1854) [10], cf. Figure 5 (with various modifications [11; 12]), the saltatory conduction observed in myelinated axons [13] is still unclear as being apparently beyond the cable theory explanation ability and other possible mechanism is searched [9; 14].