## Appendix A
\begin{split}\rho_{\mathrm{k}}^{(\omega)}(\xi\cdot\tau)=-\frac{4 at \eta_{0}}{\mathrm{d}}\cdot\frac{\mathrm{s}^{k-1}a^{k-1}\mathrm{d}^{k-1}n_{0}^{ k-1}}{\pi^{2(k-1)}\epsilon_{0}^{k-1}\epsilon_{0}^{k-1}\epsilon_{0}^{k-1} \epsilon_{0}^{k-1}}\times\\ \times\left(\sum_{\mathrm{n}_{\mathrm{k}}=1}^{\infty}\left(\frac{ \sin^{2}\left(\frac{m_{\mathrm{k}}}{2}\right)}{n_{\mathrm{k}}{}^{2}\left( \frac{1}{n_{\mathrm{k}}}+\mathrm{i}\frac{\omega}{w^{(0)}}\right)}\right) \right)^{k-1}\times\sum_{\mathrm{n}_{1}=1}^{\infty}\frac{\sin^{2}\left(\frac{ m_{\mathrm{1}}}{2}\right)}{\frac{1}{\tau_{\mathrm{n}_{1}}}+\mathrm{i}\frac{ \omega}{w^{(0)}}}\times\cos\ \left(\frac{m_{\mathrm{1}}}{\mathrm{d}}d\xi\right)\exp\left(\frac{ \mathrm{i}\omega\tau}{W^{(0)}}\right),\\ \rho_{\mathrm{k}}^{(\omega)}(\xi\cdot\tau)=-\frac{4 at\eta_{0}}{ \mathrm{d}}\cdot\frac{\mathrm{s}^{k-1}a^{k-1}\mathrm{d}^{k-1}n_{0}^{k-1}}{\pi^ {2(k-1)}\epsilon_{0}^{k-1}\epsilon_{0}^{k-1}\epsilon_{0}^{k-1}\epsilon_{0}^{ k-1}}\times\\ \times\sum_{\mathrm{n}_{1}=1}^{\infty}\sum_{\mathrm{n}_{2}=1}^{ \infty}\cdots\sum_{\mathrm{n}_{\mathrm{k}}=1}^{\infty}\left(\frac{\sin^{2} \left(\frac{m_{\mathrm{1}}}{2}\right)\sin^{2}\left(\frac{m_{\mathrm{2}}}{2} \right)\cdot\cdot\cdot\cdot\sin^{2}\left(\frac{m_{\mathrm{k}}}{2}\right)}{n_{ \mathrm{n}}{}^{2}\cdot\cdot\cdot\cdot\cdot\cdot\mathrm{n}^{2}\left(\frac{1}{n _{\mathrm{1}}}+\mathrm{i}\frac{\omega}{w^{(0)}}\right)\left(\frac{1}{\tau_{ \mathrm{n}_{2}}}+\mathrm{i}\frac{\omega}{w^{(0)}}\right)\cdot\cdot\cdot\left( \frac{1}{\tau_{\mathrm{n}_{\mathrm{k}}}}+\mathrm{i}\frac{\omega}{w^{(0)}} \right)}\right)\times\\ \times\cos\ \left(\frac{m_{\mathrm{1}}a^{2}}{\frac{1}{\tau_{\mathrm{n} _{2}}}+\mathrm{i}\frac{\omega}{w^{(0)}}}\right)\times\cos\ \left(\frac{m_{\mathrm{1}}a^{2}}{ \frac{1}{\tau_{\mathrm{n}_{1}}}+\mathrm{i}\frac{\omega}{w^{(0)}}}\right)\cdot \frac{\exp\left(2\mathrm{i}\omega\tau}{W^{(0)}}\right),\end{split} \] (A10a) \[ \begin{split}\rho_{\mathrm{k}}^{(2\omega)}(\xi\cdot\tau)=\frac{16( \mathrm{k}-1)a^{2}n_{0}}{\mathrm{d}^{2}}\cdot\frac{\mathrm{s}^{k-2}a^{k-2}a^{k -2}\ \mathrm{n}_{0}^{k-2}}{\pi^{2(k-2)}\epsilon_{0}^{k-2}\epsilon_{0}^{k-2} \epsilon_{0}^{k-2}\epsilon_{0}^{k-2}\epsilon_{0}^{k-2}\epsilon_{0}^{k-2} \epsilon_{0}^{k-2}}\times\\ \times\sum_{\mathrm{n}_{1}=1}^{\infty}\sum_{\mathrm{n}_{2}=1}^{ \infty}\cdots\sum_{\mathrm{n}_{\mathrm{k}}=1}^{\infty}\left(\frac{\sin^{2} \left(\frac{m_{\mathrm{1}}}{2}\right)\sin^{2}\left(\frac{m_{\mathrm{2}}}{2} \right)\cdot\cdot\cdot\cdot\sin^{2}\left(\frac{m_{\mathrm{k}}}{2}\right)}{n_{ \mathrm{k}}{}^{2}\cdot\cdot\cdot\cdot\cdot\mathrm{n}^{2}\left(\frac{1}{\tau_{ \mathrm{n}_{2}}}+\mathrm{i}\frac{\omega}{w^{(0)}}\right)\left(\frac{1}{\tau_{ \mathrm{n}_{\mathrm{k}}}}+\mathrm{i}\frac{\omega}{w^{(0)}}\right)\cdot\cdot \cdot\cdot\cdot\left(\frac{1}{\tau_{\mathrm{n}_{\mathrm{k}}}}+\mathrm{i}\frac{ \omega}{w^{(0)}}\right)}\times\\ \times\frac{\sin^{2}\left(\frac{m_{\mathrm{2}}}{2}\right)}{\left( \frac{1}{\tau_{\mathrm{n}_{2}}}+\mathrm{i}\frac{\omega}{w^{(0)}}\right)} \right)\times\cos\ \left(\frac{m_{\mathrm{1}}a^{2}}{\frac{1}{\tau_{\mathrm{n}_{ \mathrm{1}}}}+\mathrm{i}\frac{\omega}{w^{(0)}}}\right)\cdot\frac{\exp\left(2 \mathrm{i}\omega\tau}{W^{(0)}}\right).\end{split} \] (A10b)  
The construction of recurrent formulas of the form \(\rho_{\mathrm{k}}^{(3\omega)}(\xi\cdot\tau)\), k \(\geq\)3 via a similar method is so analytically complicated that it requires the use of a fundamentally different approach proposed in the main sections of this study. By substituting (A10a) and (A10b) into  
\[\rho^{(\omega)}(\xi\cdot\tau)=\sum_{\mathrm{k}=1}^{\infty}\gamma^{\mathrm{k}} \rho_{\mathrm{k}}^{(\omega)}(\xi\cdot\tau),\ \rho^{(2\omega)}(\xi\cdot\tau)=\sum_{\mathrm{k}=2}^{\infty}\gamma^{\mathrm{k}} \rho_{\mathrm{k}}^{(2\omega)}(\xi\cdot\tau),\] (A11)  
we obtain the following:  
\[