## 2 Materials and Methods (Theoretical Bases and Methods)
Without limiting the generality of the results, let us write the solution of Equation (11) through a linear approximation using the small dimensionless parameters \(\xi(t)=\xi_{0}\exp(\mathrm{i}\omega t)\), \(\eta(t)=\eta_{0}\exp(\mathrm{i}\omega t)\), where \(\xi_{0}=\left|\frac{(\Delta U)_{0}}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\right|\), \(\eta_{0}=\Lambda\left|\frac{(\Delta U)_{0}}{\mathrm{U}_{0}}\right|\), \((\Delta U)_{0}=\frac{1}{6}\mathrm{q}\delta_{0}(\varepsilon_{\infty}+2)\mathrm{ E}_{0}\). Then, using the approximate expressions  
\[\Delta n^{(\omega)}(t)=\xi_{0}\Big{(}\Delta n^{(\omega)}(t)\Big{)}^{(1)},\] \[\Omega^{(\omega)}(t)\approx\frac{\gamma_{0}}{2}\bigg{(}\exp(- \mathrm{X})+\frac{\exp(-\Lambda)-\exp(-\mathrm{X})}{1-\frac{\gamma_{0}}{2}} \bigg{)}=W^{q,-classic(0)},\] \[\Xi^{(\omega)}(t)\approx\
u_{0}\bigg{(}\exp(-\mathrm{X})+\frac{ \exp(-\Lambda)-\exp(-\mathrm{X})}{1-\frac{\gamma_{0}}{2}}\bigg{)}\times\xi(t )=2W^{q,-classic(0)}\times\] \[\xi_{0}\exp(\mathrm{i}\omega t),\]  
set up the equation:  
\[\frac{\mathrm{d}\Big{(}\Delta n^{(\omega)}(t)\Big{)}^{(1)}}{\mathrm{dt}}+2W^{ q,-classic;(0)}\Big{(}\Delta n^{(\omega)}(t)\Big{)}^{(1)}=\frac{2}{3}n_{0}W^{q,- classic;(0)}\times\exp(\mathrm{i}\omega t). \tag{12}\]  
Its solution, taking into account \(\Big{(}\Delta n^{(\omega)}(0)\Big{)}^{(1)}=0\), describes transient processes in the dielectric  
\[=\
u_{0}\bigg{(}\exp(-X)\text{sh}(\xi(t))+\frac{\frac{1}{2}\exp(- \Lambda)\exp(\mathfrak{n}(t))(1-\exp(-2\xi(t)))-\exp(-X)\text{sh}(\xi(t))}{1- \frac{\Lambda}{\Lambda}}\bigg{)},\]0  
In the stationary polarization mode, we have  
\[=\
u_{0}\bigg{(}\exp(-X)\text{sh}(\xi(t))+\frac{\frac{1}{2}\exp(- \Lambda)\exp(\mathfrak{n}(t))(1-\exp(-2\xi(t)))-\exp(-X)\text{sh}(\xi(t))}{1- \frac{\Lambda}{\Lambda}}\bigg{)},\]1  
When \(\omega=0\), we obtain  
\[=\
u_{0}\bigg{(}\exp(-X)\text{sh}(\xi(t))+\frac{\frac{1}{2}\exp(- \Lambda)\exp(\mathfrak{n}(t))(1-\exp(-2\xi(t)))-\exp(-X)\text{sh}(\xi(t))}{1- \frac{\Lambda}{\Lambda}}\bigg{)},\]2  
The system relaxation time \(\tau^{(\omega)}(t)=\frac{1}{2\Omega^{(\omega)}(t)}\) in the accepted approximation is \(\tau^{(\omega)}(t)\approx\frac{1}{2W^{q,-classic(0)}(\mathrm{T})}\). The solution of the kinetic Equation (11) in the form of Equations (12) and (13) is applicable in the region of weak fields \(\xi(t)\ll 1\), \(\eta(t)\ll 1\). To more strictly take into account the influence of the external perturbance parameters \(\xi(t)\), \(\eta(t)\) on the polarization of the dielectric, it is necessary to construct a generalized solution to the nonlinear kinetic equation. This approach is effective in the region of anomalously high-polarization nonlinearities, which manifest themselves in HBC in the region of ultralow temperatures (1-10 K) in weak fields (0.1-1 MV/m), and in the region of ultrahigh temperatures (550-1500 K and higher) in strong fields (10-1000 MV/m) [1; 2; 7]. The theoretical investigation of the kinetics of quantum proton relaxation and conductivity near absolute zero temperatures are a rather complicated scientific problem from a physical point of view. If, formally, we have the opportunity to extend the mathematical model of the nonlinear quantum polarization in HBC in the form of Equations (9)-(11), and in the form of [1; 2; 5] to the temperature range T = 0-1 K, then from the point of view of the fundamental physical rigor of the final results, it is extremely difficult to assert anything in this case. In [1; 2], situations with values of the dimensionless small parameter \(\xi\) of perturbation theory equal to \(\xi\) = 0.1-1 at temperatures T = 1-10 K are touched upon, but the model of these phenomena remains quasi-classical, and is not strictly quantum mechanical. At HBC temperatures near absolute zero, fundamentally new quantum kinetic phenomena appeared, associated with the interaction between proton and phonon subsystems similar to electronic superconductivity. However, at this time, these effects in HBC have not yet been explored, either theoretically or experimentally.