## 1 Introduction
### Effect of Nonlinearities on Relaxation Times
In the field of temperatures \(\mathrm{T<}\mathrm{<T_{mov}}\), \(\frac{\Lambda}{\chi}=\frac{\pi\delta_{0}\mathrm{k_{B}}\mathrm{T}\sqrt{\mathrm{ m}}}{\hbar\sqrt{2\mathrm{U}_{0}}}{<<}1\) and \(W^{(2l)}\to W^{(2l)}_{\mathrm{num}}=\frac{\gamma_{0}}{2}\langle\mathrm{D}^{(2l )}\rangle\), the Formula (31), with a limit, gives  
\[\tau_{quant,tunn}(\mathrm{T})=\frac{2}{\
u_{0}\Big{[}\mathrm{D}^{(2l)}( \mathrm{T})+\sum_{l=1}^{\infty}\frac{1}{(2l)!}\cdot D^{(2l)}(\mathrm{T}) \zeta^{2l}(\mathrm{x};\mathrm{t})\Big{]}}. \tag{47}\]  
With \(\mathrm{T<<}\mathrm{T_{cr,move}}\), \(\frac{\Lambda}{\chi(\mathrm{T})}<<1\) by transformation (10) \(D^{(2l)}(\mathrm{T})\approx\frac{\Lambda^{2l}}{\chi^{2}(\mathrm{T})\left(1- \frac{\Lambda}{\chi(\mathrm{T})}\right)}\exp(-\Lambda)\), and by assuming that \(\frac{\Lambda}{\chi(\mathrm{T})}=\frac{\eta(\mathrm{x};\mathrm{t})}{\zeta( \mathrm{x})}\), we arrive at  
\[\tau_{quant,tunn}(\mathrm{T})\approx\frac{2}{\
u_{0}\times\frac{\exp(-\Lambda )}{1-\frac{\Lambda}{\chi(\mathrm{T})}}\times\left[1+\sum_{l=1}^{\infty}\frac{1 }{(2l)!}\left(\frac{\Lambda}{\chi(\mathrm{T})}\right)^{2l}\zeta^{2l}(\mathrm{ x};\mathrm{t})\right]}=\frac{2\left(1-\frac{\Lambda}{\chi}\right)\cdot\exp(\Lambda)}{\
u_{0} \mathrm{ch}\eta(\mathrm{x};\mathrm{t})}. \tag{48}\]  
In the area of weak fields, \(\eta(\mathrm{x};\mathrm{t})=\Lambda\frac{|\Delta\mathrm{U}(\mathrm{x};\mathrm{ t})|}{\mathrm{U}_{0}}<<1\), we obtain  
\[\tau_{quant,tunn}(\mathrm{T})\approx\frac{2\left(1-\frac{\Lambda}{\chi} \right)\cdot\exp(\Lambda)}{\
u_{0}}. \tag{49}\]  
At ultra-low temperatures, when \(\frac{\Lambda}{\chi}=0\), from (49),  
\[\tau_{quant,tunn}(\mathrm{T})\approx\frac{2\cdot\exp(\Lambda)}{\
u_{0}}. \tag{50}\]  
The expressions (48) and (49) indicate weak dependence of the relaxation time on temperature in the area of tunnel passages (\(\mathrm{T<<}\mathrm{T_{mov}}\)), and Expression (50) allows us to assert that near the temperature of absolute zero, the relaxation time is a function of only the parameters of the relaxers and the parameters of the potential pattern laid down by the parameter \(\Lambda\). Formula (45), represented by \(\tau(\mathrm{T})=\frac{\frac{W^{(-)}(\mathrm{x};\mathrm{t})+W^{(+)}(\mathrm{ x})}}{\
u_{0}\Bigg{(}\exp\left(-\frac{\mathrm{U}_{0}}{\mathrm{k_{B}T}}\right) \cdot\mathrm{ch}\left(\left|\frac{\Delta\mathrm{U}(\mathrm{x};\mathrm{t})}{ \mathrm{k_{B}T}}\right|\right)+\frac{\exp(-\Lambda)\mathrm{ch}\left(\Lambda \frac{|\Delta\mathrm{U}(\mathrm{x};\mathrm{t})|}{\mathrm{k_{B}T  
where, in zero approximation by field \((\Delta\mathrm{U}=0)\), in the area of low temperatures \((\frac{\Lambda\mathrm{k_{B}T}}{\mathrm{U_{0}}}<<1)\), obviously, \(\tau^{(0)}(\mathrm{T})\approx\frac{2\left(1-\frac{\Lambda\mathrm{k_{B}T}}{ \mathrm{U_{0}}}\right)}{\mathrm{V_{0}}}\cdot\mathrm{e}^{\Lambda}\), and at ultralow temperatures, we have \(\tau^{(0)}(0)\rightarrow\frac{2}{\mathrm{V_{0}}}\mathrm{e}^{\Lambda}\), which is consistent with (49) and (50). Based on (46), by taking into account (8) and (9), it is obvious that  
\[\frac{1}{\tau(\mathrm{T})}=\left[\frac{1}{\tau^{(0)}(\mathrm{T})}+\sum_{l=1}^{ \infty}\frac{1}{(2l)!}\cdot\left(\frac{\mathrm{q}a}{2\mathrm{k_{B}T}}\right)^{ 2l}\cdot\frac{1}{\tau^{(2l)}(\mathrm{T})}\cdot\mathrm{E}^{2l}(\mathrm{x}; \mathrm{t})\right]^{-1} \tag{52}\]  
In (52),  
\[\tau^{(2l)}(\mathrm{T})=\frac{2}{\mathrm{V_{0}}}\left[\exp(-\mathrm{X})+ \frac{\left(\frac{\lambda}{\lambda}\right)^{2l}\exp(-\Lambda)-\exp(-\mathrm{X })}{1-\frac{\lambda}{\lambda}}\right]^{-1}. \tag{53}\]  
By taking \(\frac{\Lambda}{\lambda}<<1\) into account in (53), we are left with, approximately, \(\tau^{(2l)}(\mathrm{T})=\frac{2}{\mathrm{V_{0}}}\cdot\left[\frac{\left(\frac{ \Lambda}{\lambda}\right)^{2l}\exp(-\Lambda)}{1-\frac{\lambda}{\lambda}}\right] ^{-1}\), where, in the limit \(\frac{\Lambda}{\lambda}=0\) starting with the order \(2l=2\), \(\tau^{(2l)}(0)=\infty\). The exception is the case \(2l=0\), \(\tau^{(0)}(0)=\frac{2}{\mathrm{V_{0}}}\cdot\exp(\Lambda)\).