## 1 Introduction
### Nonlinear Effects Under Ion-Relaxation Polarization Conditions
 \tag{76}\]  
In (76), we introduce the following integral operators:  
\[\hat{\rm K}^{(\tau)}\Bigl{(}\Re^{(3\omega)}_{{\rm k}-1}\Bigl{(}{\rm s},\tau^{ \prime}\Bigr{)}\Bigr{)}=\exp\biggl{(}-\frac{\tau}{\tau_{\rm n}}\biggr{)}\cdot \int_{0}^{\tau}\Re^{(3\omega)}_{{\rm k}-1}\Bigl{(}{\rm s},\tau^{\prime} \Bigr{)}\exp\biggl{(}\frac{\tau^{\prime}}{\tau_{\rm n}}\Bigr{)}{\rm d}\tau^{ \prime}\longrightarrow\frac{\Re^{(3\omega)}_{{\rm k}-1}\bigl{(}{\rm s},\tau\bigr{)}}{ \frac{1}{\tau_{\rm n}}+{\rm i}\Bigl{(}3\frac{\omega}{W^{(0)}}\Bigr{)}}, \tag{76a}\]\[\hat{K}^{(\tau)}\Big{(}\exp\Big{(}\mathrm{i}\frac{\omega}{W^{(0)}} \tau^{\prime}\Big{)}\Re_{\mathrm{k-1}}^{(2\omega)}\Big{(}s,\tau^{\prime}\Big{)} \Big{)}=\exp\Big{(}-\frac{\tau}{\tau_{n}}\Big{)}\cdot\int_{0}^{\tau}\Re_{ \mathrm{k-1}}^{(2\omega)}\Big{(}s,\tau^{\prime}\Big{)}\times\] \[\times\ \exp\Big{(}\Big{(}\mathrm{i}\frac{\omega}{W^{(0)}}+\frac{1}{ \tau_{n}}\Big{)}\tau^{\prime}\Big{)}\mathrm{d}\tau^{\prime}\longrightarrow\frac {\Re_{\mathrm{k-1}}^{(2\omega)}(s,\tau)\exp\big{(}\frac{-\omega}{W^{(0)}}\tau \big{)}}{\frac{1}{\tau_{n}}+\mathrm{i}\big{(}\frac{\omega}{W^{(0)}}\big{)}} \,, \tag{76b}\] \[\hat{K}^{(\tau)}\Big{(}\Re_{\mathrm{k-m-1}}^{(2\omega)}\Big{(} \mathrm{p},\tau^{\prime}\Big{)}\Re_{\mathrm{m}}^{(\omega)}\Big{(}l,\tau^{ \prime}\Big{)}\Big{)}=\exp\Big{(}-\frac{\tau}{\tau_{n}}\Big{)}\cdot\int_{0}^ {\tau}\Re_{\mathrm{k-m-1}}^{(2\omega)}\Big{(}\mathrm{p},\tau^{\prime}\Big{)} \Re_{\mathrm{m}}^{(\omega)}\Big{(}l,\tau^{\prime}\Big{)}\exp\Big{(}\frac{\tau ^{\prime}}{\tau_{n}}\Big{)}\mathrm{d}\tau^{\prime}\longrightarrow\] \[\longrightarrow\frac{\Re_{\mathrm{k-m-1}}^{(2\omega)}(\mathrm{p}, \tau)\Re_{\mathrm{m}}^{(\omega)}(l,\tau)}{\frac{1}{\tau_{n}}+\mathrm{i}\big{(} \frac{\omega}{W^{(0)}}\big{)}}\,,\] (76c) \[\hat{K}^{(\tau)}\Big{(}\Re_{\mathrm{k-m-1}}^{(\omega)}\Big{(} \mathrm{p},\tau^{\prime}\Big{)}\Re_{\mathrm{m}}^{(2\omega)}\Big{(}l,\tau^{ \prime}\Big{)}\Big{)}=\exp\Big{(}-\frac{\tau}{\tau_{n}}\Big{)}\cdot\int_{0}^ {\tau}\Re_{\mathrm{k-m-1}}^{(\omega)}\Big{(}\mathrm{p},\tau^{\prime}\Big{)} \Re_{\mathrm{m}}^{(2\omega)}\Big{(}l,\tau^{\prime}\Big{)}\exp\Big{(}\frac{\tau ^{\prime}}{\tau_{n}}\Big{)}\mathrm{d}\tau^{\prime}\longrightarrow\] \[\longrightarrow\frac{\Re_{\mathrm{k-m-1}}^{(\mathrm{p},\tau)} \Re_{\mathrm{m}}^{(2\omega)}(l,\tau)}{\frac{1}{\tau_{n}}+\mathrm{i}\big{(} \frac{\omega}{W^{(0)}}\big{)}}\,. \tag{76d}\]  
The contribution to the amplitude \(\Re_{\mathrm{k}}^{(3\omega)}(\mathrm{n},\tau)\) of the second term of the recurrence relation (76) can be obtained by considering (75):  
\[\Re_{\mathrm{k-1}}^{(2\omega)}(\mathrm{s},\tau)=(\mathrm{k}-2)\bigg{(}\frac{4 a}{\mathrm{d}}\bigg{)}^{2}n_{0}\bigg{(}\frac{8n_{0}\phi_{1}}{\pi^{2}}\bigg{)}^{ \mathrm{k}-3}\cdot\Lambda_{0}^{\mathrm{k}-3}\cdot\mathrm{s}^{2}\cos^{2}\!\big{(} \frac{7\mathrm{s}}{2}\big{)}\times\wp_{2}(\mathrm{s})\times\frac{\exp\!\left( 2\mathrm{i}\frac{\omega}{W^{(0)}}\tau\right)}{\frac{1}{\tau_{n}}+\mathrm{i} \Big{(}2\frac{\omega}{W^{(0)}}\big{)}}, \tag{77}\]  
where \(\wp_{2}(\mathrm{s})=\sum_{\mathrm{m}=1}^{\infty}\frac{\sin^{2}\big{(}\frac{ \mathrm{m}}{2}\big{)}}{(\mathrm{s}^{2}-\mathrm{m}^{2})\big{(}\frac{1}{\tau_{m} }+\mathrm{i}\frac{\omega}{W^{(0)}}\big{)}}\). Substituting (77) into the second summand from (76) yields  
\[\Big{[}\Re_{\mathrm{k}}^{(3\omega)}(\mathrm{n},\tau)\Big{]}_{2}=-(\mathrm{k}-2 )\bigg{(}\frac{4a}{\mathrm{d}}\bigg{)}^{3}n_{0}\bigg{(}\frac{8n_{0}\phi_{1}}{ \pi^{2}}\bigg{)}^{\mathrm{k}-3}\cdot\Lambda_{0}^{\mathrm{k}-3}\cdot\mathrm{n}^ {2}\times\wp_{3}(\mathrm{n})\times\frac{\exp\!\left(3\mathrm{i}\frac{\omega }{W^{(0)}}\tau\right)}{\frac{1}{\tau_{n}}+\mathrm{i}\Big{(}3\frac{\omega}{W^{( 0)}}\big{)}}. \tag{78}\]  
Here, the designation \(\wp_{3}(\mathrm{n})=\sum_{\mathrm{m}=1}^{\infty}\left\{\frac{\mathrm{m}^{2} \wp_{2}(\mathrm{m})\cos^{2}\!\big{(}\frac{\mathrm{m}}{2}\big{)}\sin^{2}\!