## 6 Conclusions
This can start from the third order via the decomposition parameter \(\gamma=\frac{\zeta_{0}W^{(1)}}{W^{(0)}}\), defined under the condition \(\zeta_{0}=\frac{\mathrm{q}E_{0}a}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}<<1\) and \(\frac{W^{(1)}}{W^{(0)}}\leq 1\). 5. The solution schemes of the nonlinear equations of the phenomenological model in the k-th approximation of the perturbation theory via the small parameter \(\gamma\) were presented and analyzed (expressions (63)-(67)) using decompositions of the volumetric charge density \(\rho_{k}(k;\tau)\) on \(0\leq k\leq\frac{\mathrm{d}}{\mathrm{d}}\) with respect to the Fourier dimension: \(\rho_{k}(k;\tau)=\sum_{\alpha=1}^{\infty}\Re_{k}(n,\tau)\cos(\frac{\gamma m \alpha}{\mathrm{d}}\xi)\), \(\Re_{k}(n,\tau)=\frac{2a}{\mathrm{d}}\int_{0}^{\frac{\mathrm{d}}{\mathrm{d}}} \rho_{k}(k;\tau)\cos(\frac{\gamma m\alpha}{\mathrm{d}}\xi)\mathrm{d}\xi\) (see expressions (68), (69), and (70)). The asymptotic recurrence expression for complex amplitudes \(\Re_{k}^{(\gamma\omega)}(n,\tau)\) of the k-order relaxation modes of perturbation theory was constructed; multiple frequencies of \"r \(\omega\)\" in the stationary polarization mode (Formula (73)) and, based on (62), according to \(\rho_{k}^{(\gamma\omega)}(x;t)=\sum_{\alpha=1}^{\infty}\Re_{k}^{(\gamma\omega)} (n,\tau)(n,\tau)\cos(\frac{\gamma m\alpha}{\mathrm{d}})\) were examined, and the decomposition of the volumetric charge density \(\rho(x;t)\) via the frequency harmonics of the alternating field was obtained (85), where \(\rho^{(\omega)}\sim E_{\rm pol}(t)\) in (A12a) is the first, \(\rho^{(2\omega)}\sim\left(E_{\rm pol}(t)\right)^{2}\) (A12b) is the second, and \(\rho^{(3\omega)}\sim\left(E_{\rm pol}(t)\right)^{3}\) (90) is the third approximation via the polarization of the (external) alternating electric field. * Using quasi-classical kinetic theory methods, it was established that in the dielectric and under the action of an alternating electric field \(E_{\rm pol}(t)\), relaxation modes are generated, and they are multiples of only odd frequencies \(((2\lambda+1)\omega)\). The expression for complex dielectric permittivity (CDP) \(\hat{\varepsilon}^{(\Omega)}\) is written based on the set of frequencies \(\Omega=\{\omega;2\omega;3\omega;\ldots;2\lambda\omega;(2\lambda+1)\omega;\ldots\}\) in the form of a series decomposition; this is carried out relative to the even frequency harmonics of the alternating field \(\hat{\varepsilon}^{(\Omega_{2\lambda+1})}\sim e^{2i\lambda\omega t}\) (expression (98)). Moreover, function \(\hat{\varepsilon}^{(\Omega)}\) is presented in the form of an infinite power series with even tension degrees of \(E_{\rm pol}^{2\lambda}(t)\); decomposition coefficients \(\hat{\varepsilon}_{2\lambda}^{(\Omega_{2\lambda+1})}=\frac{\hat{\varepsilon} ^{(\Omega_{2\lambda+1})}}{E_{\rm pol}^{2\lambda}(t)}\) are used (expression (97)). The generalized dispersion relations of the forms \(\left\langle\,\mathrm{Re}\left[\stackrel{{-\! (\Omega)}}{{\rm E} _{\rm pol}}(t)\right]\cdot\mathrm{Re}\left[\stackrel{{-\! (\Omega)} }{{j}}(t)\right]\right\rangle\), \(\left\langle\,\mathrm{Im}\left[\stackrel{{-\! (\Omega)}}{{\rm E} _{\rm pol}}(t)\right]\cdot\mathrm{Im}\left[\stackrel{{-\! ( \Omega)}}{{j}}(t)\right]\right\rangle\), \(\left\langle\,\mathrm{Im}\left[\stackrel{{-\! (\Omega)}}{{\rm E} _{\rm pol}}(t)\right]\cdot\mathrm{Re}\left[\stackrel{{-\! ( \Omega)}}{{j}}(t)\right]\right\rangle\), and \(\left\langle\,\mathrm{Re}\left[\stackrel{{-\! (\Omega)}}{{\rm E} _{\rm pol}}(t)\right]\cdot\mathrm{Im}\left[\stackrel{{-\! (\Omega)} }{{j}}(t)\right]\right\rangle\) can be observed in Formulas (113a)-(113d). Here, in the particular case of \(\Omega=\{\Omega_{1};\widetilde{\Omega}_{3}\}\), the expressions known from the electrodynamics of continuous media are obtained. The function \(\hat{\varepsilon}^{(\Omega_{1},\Omega_{3})}(\omega;T)\) is obtained in its analytical form (expression (108)).