## 1 Introduction
### Nonlinear Effects Under Ion-Relaxation Polarization Conditions
\begin{split}\Bigl{[}\Re_{\text{k}}^{(3\omega)}(\text{n},\tau) \Bigr{]}_{1,g}=-\Bigl{(}\frac{4a}{\text{d}}\Bigr{)}^{3}n_{0}\Bigl{(}\frac{8n_{0 }\phi_{1}}{\pi^{2}}\Bigr{)}^{\text{k}-3}\Lambda_{0}^{g-3}\Lambda_{1}\Lambda_{ 2}^{k-g-1}\frac{(g-1)(g-2)}{2}\times\\ \times\sin^{2}\bigl{(}\frac{2\pi}{2}\bigr{)}\times\frac{\exp \bigl{(}3i\frac{\omega}{W^{(0)}}\tau\bigr{)}}{\frac{1}{\tau_{\text{n}}}+i \bigl{(}3\frac{\omega}{W^{(0)}}\bigr{)}}.\end{split}  \tag{81}\]  
In (81), the sum of the series \(\Lambda_{1}=\sum_{\text{p}=1}^{\infty}\Biggl{\{}\frac{\wp_{3}(\text{p})\cdot \sin^{2}\bigl{(}\frac{\text{mp}}{2}\bigr{)}}{\frac{1}{\tau_{\text{p}}}+i\bigl{(} 3\frac{\omega}{W^{(0)}}\bigr{)}}\Biggr{\}}\), by virtue of (78a), reduces to the form  
\[\Lambda_{1}=\sum_{\text{p}=1}^{\infty}\sum_{\text{m}=1}^{\infty}\sum_{\text{s }=1}^{\infty}\Biggl{\{}\frac{\text{m}^{2}\cos^{2}\bigl{(}\frac{\text{mp}}{2} \bigr{)}\text{sm}^{2}\bigl{(}\frac{\text{mp}}{2}\bigr{)}}{(\text{p}^{2}-\text{ m}^{2})(\text{m}^{2}-\text{s}^{2})\bigl{(}\frac{1}{\tau_{\text{s}}}+i\bigl{(} \frac{\omega}{W^{(0)}}\bigr{)}\bigl{)}\bigl{(}\frac{1}{\tau_{\text{m}}}+i \bigl{(}2\frac{\omega}{W^{(0)}}\bigr{)}\bigr{)}\bigl{(}\frac{1}{\tau_{\text{p}} }+i\bigl{(}3\frac{\omega}{W^{(0)}}\bigr{)}\bigr{)}}\Biggr{\}}.\] (81a) In ( 81 ), the sum of the series is also included: \[\Lambda_{2}=\sum_{\text{p}=1}^{\infty}\Biggl{\{}\frac{\sin^{2}\bigl{(}\frac{ \text{mp}}{2}\bigr{)}}{\frac{1}{\tau_{\text{p}}}+i\Bigl{(}3\frac{\omega}{W^{(0) }}\Bigr{)}}\Biggr{\}}.\] (81b) The total contribution of Expression ( 81 ) to the complex amplitude \[\Bigl{[}\Re_{\text{k}}^{(3\omega)}(\text{n},\tau)\Bigr{]}\] in the k \[\geq 4\] approximation is determined by summing the elements \[\Bigl{[}\Re_{\text{k}}^{(3\omega)}(\text{n},\tau)\Bigr{]}_{1,g}\] over g \[\text{ from g = 3 to g = k - 1: \Bigl{[}\Re_{\text{k}}^{(3\omega)}(\text{n},\tau)\Bigr{]}_{1}=\sum_{\text{g }=3}^{k-1}\Bigl{[}\Re_{\text{k}}^{(3\omega)}(\text{n},\tau)\Bigr{]}_{l,g}.\] Then, \[\Bigl{[}\Re_{\text{k}}^{(3\omega)}(\text{n},\tau)\Bigr{]}_{1}=-\Bigl{(} \frac{4a}{\text{d}}\Bigr{)}^{3}n_{0}\Bigl{(}\frac{8n_{0}\phi_{1}}{\pi^{2}} \Bigr{)}^{\text{k}-3}\cdot\Lambda_{1}\cdot\sin^{2}\bigl{(}\frac{\text{mp}}{2} \bigr{)}\times\sum_{g=3}^{k-1}\frac{(g-1)(g-2)}{2}\Lambda_{0}^{g-3}\cdot\Lambda_{ 2}^{\text{k}-g-1}\times\] (82) \[\times\frac{\exp\bigl{(}3i\frac{\omega}{W^{(0)}}\tau\bigr{)}}{\frac{1 }{\tau_{\text{n}}}+i\bigl{(}3\frac{\omega}{W^{(0)}}\bigr{)}}.\] The full expression for the complex amplitude \(\Bigl{[}\Re_{\text{k}}^{(3\omega)}(\text{n},\tau)\Bigr{]}=\Bigl{[}\Re_{\text{k}}^{(3 \omega)}(\text{n},\tau)\Bigr{]}_{2,3}+\Bigl{[}\Re_{\text{k}}^{(3\omega)}(\text{n}, \tau)\Bigr{]}_{1}\) takes the following form:\[ \begin{array}{c}\Big{[}\Re_{k}^{(3\omega)}(n,\tau)\Big{]}=-\Big{(}\frac{4a}{d} \Big{)}^{3}n_{0}\Big{(}\frac{8n_{0}\phi_{1}}{\pi^{2}}\Big{)}^{k-3}\times\\ \times\Big{\{}\frac{(k-1)(k-2)}{2}\Lambda_{0}^{k-3}\cdot n^{2}\wp_{3}(n)+\Lambda _{1}\sin^{2}(\frac{\gamma m}{2})\times\sum_{g-3}^{k-1}\frac{(g-1)(g-2)}{2} \Lambda_{0}^{g-3}\cdot\Lambda_{2}^{k-g-1}\Big{\}}\times\\ \times\frac{\exp(3i\frac{\omega}{\pi^{2}(0)}\tau)}{\frac{1}{\pi_{2}}+i\Big{(} \frac{3}{\omega^{0}(0)}\tau)}.\end{array}  \tag{83}\]  
Expression (83), in a special case, at k = 3, agrees with (A9f).