## 3 Studies of the Nonlinear Kinetic Equation
\begin{gathered}\Re_{k}(n,\tau)=\frac{8n_{0}\varphi_{1}}{\tau^{2 }}\sum\limits_{s=1}^{\infty}\bigg{\{}\frac{\sin^{2}\big{(}\frac{\pi s}{2} \big{)}\sin^{2}\big{(}\frac{\pi m}{2}\big{)}}{s^{2}}\times\Big{[}\exp\!\Big{(} -\frac{\tau}{\tau_{n}}\Big{)}\cdot\int_{0}^{\tau}\Re_{k-1}(s,\tau)\exp\Big{(} \frac{\tau}{\tau_{n}}\Big{)}d\tau\Big{]}\bigg{\}}-\frac{4a}{d}.\\ \cdot\sum\limits_{s=1}^{\infty}\bigg{\{}\frac{n^{2}\cos^{2}\big{(} \frac{\pi m}{2}\big{)}\sin^{2}\big{(}\frac{\pi s}{2}\big{)}}{n^{2}-s^{2}} \times\Big{[}\exp\!\Big{(}-\frac{\tau}{\tau_{n}}\Big{)}\cdot\int_{0}^{\tau} \Re_{k-1}(s,\tau)\cdot\exp\Big{(}\Big{(}\frac{\omega}{W^{(0)}}+\frac{1}{\tau_{n }}\Big{)}\tau\Big{)}d\tau\Big{]}\bigg{\}}+\\ +\frac{8\varphi_{1}}{\pi^{2}}\sum\limits_{m=1}^{k-2}\sum\limits_{p =1}^{\infty}\sum\limits_{l=1}^{\infty}\bigg{\{}\frac{n^{2}\cos^{2}\big{(} \frac{\pi m}{2}\big{)}\sin^{2}\big{(}\frac{\pi m}{2}\big{)}\sin^{2}\big{(} \frac{\pi l}{l}\big{)}}\times\\ \times\Big{[}\exp\!\Big{(}-\frac{\tau}{\tau_{n}}\Big{)}\times\int_ {0}^{\tau}\Re_{k-m-1}(p,\tau)\Re_{m}(l,\tau)\cdot\exp\Big{(}\frac{\tau}{\tau_{ n}}\Big{)}d\tau\Big{]}\bigg{\}}.\end{gathered}  \tag{17}\]  
Since the generation of relaxation modes with complex amplitudes \(\Re_{k}^{(\omega)}(n,\tau)\) begins with the first order of perturbation theory [1]  
from the first term of (17) it is not difficult to see  
\[\Re_{2}^{(\omega)}(n,\tau)=-\frac{4 an_{0}}{d}\times\frac{8n_{0}\varphi_{1}}{ \pi^{2}}\times\sin^{2}\!\Big{(}\frac{\tau n}{2}\Big{)}\times\!\sum_{s=1}^{ \infty}\frac{\sin^{2}\big{(}\frac{\pi s}{2}\big{)}}{s^{2}\Big{(}\frac{1}{\tau_{ s}}+\mathrm{i}\frac{\omega}{W^{(0)}}\Big{)}}\times\frac{\exp\!\Big{(} \mathrm{i}\frac{\omega}{W^{(0)}}\tau\Big{)}}{\frac{1}{\tau_{n}}+\mathrm{i} \frac{\omega}{W^{(0)}}},\]  
\[\Re_{3}^{(\omega)}(n,\tau)=-\frac{4 an_{0}}{d}\times\bigg{(}\frac{8n_{0}\varphi_{1 }}{\pi^{2}}\bigg{)}^{2}\times\sin^{2}\!\Big{(}\frac{\tau m}{2}\Big{)}\times \left(\sum_{s=1}^{\infty}\frac{\sin^{2}\big{(}\frac{\pi s}{2}\big{)}}{s^{2}\Big{(} \frac{1}{\tau_{s}}+\mathrm{i}\frac{\omega}{W^{(0)}}\Big{)}}\right)^{2}\times \frac{\exp\!\Big{(}\mathrm{i}\frac{\omega}{W^{(0)}}\tau\Big{)}}{\frac{1}{\tau_{ n}}+\mathrm{i}\frac{\omega}{W^{(0)}}},\]  
where we get the recurring formula  
\[\Re_{k}^{(\omega)}(n,\tau)=-\frac{4 an_{0}}{d}\times\bigg{(}\frac{8n_{0}\varphi_{1 }}{\pi^{2}}\bigg{)}^{k-1}\times\sin^{2}\!\Big{(}\frac{\tau n}{2}\Big{)} \times\lambda_{0}^{k-1}\times\frac{\exp\!\Big{(}\mathrm{i}\frac{\omega}{W^{(0)}} \tau\Big{)}}{\frac{1}{\tau_{n}}+\mathrm{i}\frac{\omega}{W^{(0)}}}. \tag{17a}\]To develop a recurring expression for the complex amplitudes \(\Re_{k}^{(r\omega)}(\mathfrak{n},\tau)\) of higher orders (multiples of frequency \(r\omega\)), we rewrite the expression (17) as  
\[ \begin{array}{c}\Re_{k}^{(r\omega)}(\mathfrak{n},\tau)=\frac{8n_{0}\varphi_{1 }}{\pi^{2}}\sum\limits_{\mathfrak{s}=1}^{\infty}\biggl{\{}\frac{\sin^{2}(\frac {\varpi}{2})\sin^{2}(\frac{\varpi}{2})}{\mathfrak{s}^{2}}\times\left[\hat{ \mathbb{K}}^{(\tau)}\Bigl{(}\mathfrak{R}_{\mathrm{k-1}}^{(r\omega)}(\mathfrak{ s},\tau)\Bigr{)}\right]\biggr{\}}-\\ -\frac{4a}{\mathrm{d}}\cdot\sum\
olimits_{\mathfrak{s}=1}^{\infty}\biggl{\{} \frac{\mathrm{n}^{2}\cos^{2}(\frac{\varpi}{2})\sin^{2}(\frac{\varpi}{2})}{ \mathrm{n}^{2}-\mathfrak{s}^{2}}\times\left[\hat{\mathbb{K}}^{(\tau)}\Bigl{(} \exp\Bigl{(}\mathrm{i}\frac{\omega}{W^{(0)}}\tau\Bigr{)}\mathfrak{R}_{\mathrm{ k-1}}^{((r-1)\omega)}(\mathfrak{s},\tau)\Bigr{)}\right]\biggr{\}}+\\ +\frac{8\varphi_{1}}{\pi^{2}}\sum\limits_{\mathfrak{m}=1}^{\infty}\sum \limits_{\mathfrak{p}=1}^{\infty}\sum\limits_{\mathfrak{m}=1}^{\infty}\sum \limits_{\mathfrak{m}=1}^{\infty}\frac{\mathrm{n}^{2}\cos^{2}(\frac{\varpi}{2 })\sin^{2}(\frac{\varpi}{2})}{\mathrm{l}^{2}(\mathrm{n}^{2}-\mathrm{p}^{2})} \times\\ \times\Bigl{[}\hat{\mathbb{K}}^{(\tau)}\Bigl{(}\mathfrak{R}_{\mathrm{k-m}-1}^{( (r-\tau)\omega)}(\mathfrak{p},\tau)\Re_{\mathrm{m}}^{(f\omega)}(l,\tau)\Bigr{)} \Bigr{]}.\end{array}