## 1 Introduction
### Nonlinear Effects Under Ion-Relaxation Polarization Conditions
 \tag{74b}\]  
According to (72),  
\[\Re^{(\omega)}_{{\rm k}-1}({\rm s},\tau)=-\frac{4 an_{0}}{{\rm d}}\times\biggl{(} \frac{8n_{0}\phi_{1}}{\pi^{2}}\biggr{)}^{{\rm k}-2}\cdot\Lambda^{{\rm k}-2}_{0} \sin^{2}\Bigl{(}\frac{\pi{\rm s}}{2}\Bigr{)}\times\frac{\exp\Bigl{(}{\rm i} \frac{\omega}{W^{(0)}}\tau\Bigr{)}}{\frac{1}{\tau_{\rm s}}+{\rm i}\frac{\omega }{W^{(0)}}},\]  
\[\Re^{(3\omega)}_{{\rm k}-{\rm m}-1}({\rm p},\tau)=-\frac{4 an_{0}}{{\rm d}} \times\biggl{(}\frac{8n_{0}\phi_{1}}{\pi^{2}}\biggr{)}^{{\rm k}-{\rm m}-2} \cdot\Lambda^{{\rm k}-{\rm m}-2}_{0}\sin^{2}\Bigl{(}\frac{\pi{\rm p}}{2} \Bigr{)}\times\frac{\exp\Bigl{(}{\rm i}\frac{\omega}{W^{(0)}}\tau\Bigr{)}}{ \frac{1}{\tau_{\rm n}}+{\rm i}\frac{\omega}{W^{(0)}}},\]  
\[\Re^{(\omega)}_{{\rm k}}(l,\tau)=-\frac{4 an_{0}}{{\rm d}}\times\biggl{(}\frac{8 n_{0}\phi_{1}}{\pi^{2}}\biggr{)}^{{\rm m}-1}\cdot\Lambda^{{\rm m}-1}_{0}\sin^{2} \Bigl{(}\frac{\pi l}{2}\Bigr{)}\times\frac{\exp\Bigl{(}{\rm i}\frac{\omega}{W^ {(0)}}\tau\Bigr{)}}{\frac{1}{\tau_{\rm l}}+{\rm i}\frac{\omega}{W^{(0)}}}.\]  
From (74), we obtain the following recurrence formula:  
\[\Re^{(2\omega)}_{{\rm k}}({\rm n},\tau)=({\rm k}-1)\biggl{(}\frac{4a}{{\rm d}} \biggr{)}^{2}n_{0}\biggl{(}\frac{8n_{0}\phi_{1}}{\pi^{2}}\biggr{)}^{{\rm k}-2} \cdot\Lambda^{{\rm k}-2}_{0}\cdot{\rm n}^{2}\cos^{2}\Bigl{(}\frac{\pi{\rm m}}{ 2}\Bigr{)}\times\wp_{2}({\rm n})\times\frac{\exp\Bigl{(}2{\rm i}\frac{\omega}{W ^{(0)}}\tau\Bigr{)}}{\frac{1}{\tau_{\rm n}}+{\rm i}\Bigl{(}2\frac{\omega}{W^ {(0)}}\Bigr{)}}, \tag{75}\]  
\[\text{Here, }\wp_{2}({\rm n})=\sum_{{\rm S}=1}^{\infty}\frac{\sin^{2}\bigl{(} \frac{\pi{\rm s}}{2}\bigr{)}}{(\frac{1}{\tau_{\rm s}}+{\rm i}\frac{\omega}{W^ {(0)}}\bigr{)}}.\]  
Since the generation of relaxation modes with amplitudes \(\Re^{(3\omega)}_{{\rm k}}({\rm n},\tau)\) starts from the third order of perturbation theory, according to (73),  
\[ \begin{split}\Re^{(3\omega)}_{{\rm k}}({\rm n},\tau)=\frac{8n_{0} \phi_{1}}{\pi^{2}}\sum_{{\rm S}=1}^{\infty}\biggl{\{}\frac{\sin^{2}\bigl{(} \frac{\pi{\rm s}}{2}\bigr{)}\sin^{2}\bigl{(}\frac{\pi{\rm s}}{2}\bigr{)}}{s^{2}} \times\Bigl{[}\hat{\rm K}^{(\tau)}\Bigl{(}+\Re^{(3\omega)}_{{\rm k}-1}\Bigl{(}{ \rm s},\tau^{\prime}\Bigr{)}\Bigr{)}\Bigr{]}\biggr{\}}-\\ -\frac{4a}{{\rm d}}\cdot\sum_{{\rm S}=1}^{\infty}\biggl{\{}\frac{n^{2} \cos^{2}\bigl{(}\frac{\pi{\rm s}}{2}\bigr{)}\sin^{2}\bigl{(}\frac{\pi{\rm s}}{2} \bigr{)}}{n^{2}-s^{2}}\times\Bigl{[}\hat{\rm K}^{(\tau)}\Bigl{(}\exp\Bigl{(}{ \rm i}\frac{\omega}{W^{(0)}}\tau^{\prime}\Bigr{)}\Re^{(2\omega)}_{{\rm k}-1} \Bigl{(}{\rm s},\tau^{\prime}\Bigr{)}\Bigr{)}\Bigr{]}\biggr{\}}+\\ +\frac{8\phi_{1}}{\pi^{2}}\sum_{{\rm m}=1}^{{\rm k}-2}\sum_{{\rm p}=1}^ {\infty}\sum_{l=1}^{\infty}\biggl{\{}\frac{n^{2}\cos^{2}\bigl{(}\frac{\pi{\rm s}}{2} \bigr{)}\sin^{2}\bigl{(}\frac{\pi{\rm p}}{2}\bigr{)}}{\hat{\rm I}}^{(2-{\rm p} ^{2})}\cdot{\rm K}^{(\tau)}\Bigl{[}+\Re^{(2\omega)}_{{\rm k}-{\rm m}-1} \Bigl{(}{\rm p},\tau^{\prime}\Bigr{)}\Re^{(\omega)}_{{\rm m}}\Bigl{(}l,\tau^{ \prime}\Bigr{)}+\\ +\Re^{(\omega)}_{{\rm k}-{\rm m}-1}\Bigl{(}{\rm p},\tau^{\prime} \Bigr{)}\Re^{(2\omega)}_{{\rm m}}\Bigl{(}l,\tau^{\prime}\Bigr{)}\Bigr{]} \biggr{\}}.\end{split}