## 1 Introduction
### Investigation of the Generalized Nonlinear Kinetic Equation of Ion Relaxation
 \tag{32}\]  
By neglecting the summand of the order \(\rho^{(m)}\cdot\rho(x;t)\) in (32) and introducing the notation \(\gamma^{(2l+1)}=\frac{E_{0}\mu_{m\mathrm{ob}}^{(2l+1)}a}{D_{diff}^{(2l)}},\phi= \frac{q\mu}{\varepsilon_{0}\varepsilon_{0}\varepsilon_{0}},\rho^{(2l+1)}=n_{0} \phi\gamma^{(2l+1)}\), \(\xi=\frac{x}{a}\), we obtain \[\begin{split}\sum\limits_{k=0}^{\infty}\zeta_{0}^{k}.\frac{\partial \rho^{(k)}}{\partial t}=\sum\limits_{m=0}^{\infty}\sum\limits_{l=0}^{\infty}W^{ (2l)}\bigg{\{}\frac{1}{(2l)!}.\frac{\partial^{2}}{\partial\xi^{2}}\Big{(}\rho^ {(m)}\mathrm{z}^{2l}\Big{)}-\frac{1}{(2l+1)!}\gamma^{(2l+1)}\frac{\partial}{ \partial\xi}\Big{(}\rho^{(m)}\mathrm{z}^{2l+1}\Big{)}\bigg{\}}\zeta_{0}^{2l+m} +\\ +\sum_{m=0}^  
and \(m=1\) define the function \(\widehat{\rho}^{(7)}=\rho^{(7)}\zeta_{0}^{7}\), with \(n_{0}\cdot\gamma^{(1)}\cdot z\), \(\rho^{(5)}\zeta_{0}^{7}=\rho^{(5)}\zeta_{0}^{2}\), \(n_{0}\cdot\gamma^{(3)}\cdot z^{3}\cdot\zeta_{0}^{2}\), \(\rho^{(3)}\zeta_{0}^{7}=\rho^{(3)}\zeta_{0}^{4}\), \(\gamma^{(5)}\cdot z^{5}\cdot\zeta_{0}^{4}\) and \(\rho^{(1)}\zeta_{0}^{7}=\rho^{(1)}\zeta_{0}^{6}\), \(\gamma^{(7)}\cdot z^{7}\cdot\zeta_{0}^{6}\), etc.  
In the even-order approximation \(k=2l+m=2s\) using the parameter \(\zeta_{0}\),  
\(l=0\),\(m=2s,l=1,\ m=2(s-1),l=2\),\(m=2(s-2),l=2\), \(m=2(s-3)\),  
\(l=4,\ m=2(s-4),\ldots,l=1,\ m=2(s-1),\ldots,l=s,\ m=0\),  
the function \(\widehat{\rho}^{(2s)}=\rho^{(2s)}\zeta_{0}^{2s}\) is defined by  
\(n_{0}\cdot\gamma^{(1)}\cdot z\),  
\(\rho^{(2(s-1))}\zeta_{0}^{2s}=\widehat{\rho}^{(2(s-1))}\zeta_{0}^{2}\), \(n_{0}\cdot\gamma^{(3)}\cdot z^{3}\cdot\zeta_{0}^{2}\), \(\rho^{(2(s-2))}\zeta_{0}^{2s}=\widehat{\rho}^{(2(s-2))}\zeta_{0}^{4}\), \(\gamma^{(5)}\cdot z^{5}\cdot\zeta_{0}^{4}\)  
\(\rho^{(2(s-3))}\zeta_{0}^{2s}=\widehat{\rho}^{(2(s-3))}\zeta_{0}^{6}\), \(\gamma^{(7)}\cdot z^{7}\cdot\zeta_{0}^{2(s-4)}\zeta_{0}^{2s}=\widehat{\rho}^{ \wedge(2(s-4))}\zeta_{0}^{8}\), \(\gamma^{(9)}\cdot z^{9}\cdot\zeta_{0}^{8}\ldots\),  
\(\rho^{(2(s-l))}\zeta_{0}^{2s}=\widehat{\rho}^{(2(s-1))}\zeta_{0}^{2l}\), \(\gamma^{(2l+1)}\cdot z^{2l+1}\cdot\zeta_{0}^{2l}\), \(\ldots\),  
\(\rho^{(0)}\zeta_{0}^{2s}=\widehat{\rho}^{(1)}\zeta_{0}^{2s-1}\), \(\gamma^{(2s+1)}\cdot z^{2s+1}\cdot\zeta_{0}^{2s}\).  
In the odd-order approximation \(k=2l+m=2s+1\) performed according to the parameter  
\(\zeta_{0}\),  
\(l=0\),\(m=2s+1\),\(l=1\), \(m=2(s-1)+1\), \(l=2\), \(m=2(s-2)+1\), \(m=2(s-3)\), \(l=4\), \(m=2(s-4)\),..., \(l=l\), \(m=2(s-l)\),..., \(l=s\), and \(m=0\), the function \(\widehat{\rho}^{(2s+1)}=\rho^{(2s+1)}\zeta_{0}^{2s+1}\) is defined with  
\(n_{0}\cdot\gamma^{(1)}\cdot z\),  
\(\rho^{(2(s-l)+1)}\zeta_{0}^{2s+1}=\widehat{\rho}^{(2(s-l)+1)}\zeta_{0}^{2}\), \(\gamma^{(3)}\cdot z^{3}\cdot\zeta_{0}^{2}\), \(\rho^{(2(s-2)+1)}\zeta_{0}^{2s+1}=\widehat{\rho}^{\wedge(2(s-2)+1)}\zeta_{0}^{ 4}\), \(\gamma^{(5)}\cdot z^{5}\cdot\zeta_{0}^{4}\),  
\(\rho^{(2(s-3)+1)}\zeta_{0}^{2s}=\widehat{\rho}^{(2(s-3)+1)}\zeta_{0}^{6}\), \(\gamma^{(7)}\cdot z^{7}\cdot\zeta_{0}^{6}\), \(\rho^{(2(s-4)+1)}\zeta_{0}^{2s+1}=\widehat{\rho}^{(2(s-4)+1)}\zeta_{0}^{8}\), \(\gamma^{(9)}\cdot z^{9}\cdot\zeta_{0}^{8}\), \(\ldots\),  
\(\rho^{(2(s-l)+1)}\zeta_{0}^{2s+1}=\widehat{\rho}^{(2(s-l)+1)}\zeta_{0}^{2l}\), \(\gamma^{(2l+1)}\cdot z^{2l+1}\cdot\zeta_{0}^{2l}\), \(\ldots\),  
\(\rho^{(0)}\zeta_{0}^{2s+1}=\widehat{\rho}^{\wedge(1)}\zeta_{0}^{2s}\), \(\gamma^{(2s+1)}\cdot z^{(2s+1)}\cdot\zeta_{0}^{2s}\).  
For a complete description of the solution scheme of the kinetic Equation (28), we represent (22) and (23) in the following form:  
\[\frac{\partial z\Big{(}\xi_{i}\tau^{(0)}\Big{)}}{\partial\xi}=\phi\rho\Big{(} \xi_{i}\tau^{(0)}\Big{)},\ \int_{0}^{4/s}z\Big{(}\xi_{i}\tau^{(0)}\Big{)}\,\mathrm{d}\xi=\frac{\mathrm{d}} {a}\cdot\exp\Bigg{(}\frac{\mathrm{i}\omega\tau^{(0)}}{W^{(0)}}\Bigg{)}. \tag{44}\]  
The direct implementation of this scheme in the form of the analytical functions \(\widehat{\rho}^{(2s)}\Big{(}\xi_{i}\tau^{(0)}\Big{)}=\rho^{(2s)}\Big{(}\xi_{i} \tau^{(0)}\Big{)}\cdot\zeta_{0}^{2s}\) and \(\widehat{\rho}^{(2s+1)}\Big{(}\xi_{i}\tau^{(0)}\Big{)}=\rho^{(2s+1)}\Big{(}\xi _{i}\tau^{(0)}\Big{)}\cdot\zeta_{0}^{2s+1}\) is outside the scope of this work and will be addressed in future research.