## 2 Theoretical Model
Every neighboring layers are assumed to be perfectly bonded. Therefore, the eigenwaves in all layers will have the same longitudinal wave vector \(\zeta\) and circular frequency \(\omega\):  
\[ \begin{split}\Phi_{m}&=f_{m}(r)\cos(n\phi)\cos( \omega+\zeta x),\\ H_{m}&=h_{m}(r)\sin(n0)\sin(\omega t+\zeta x),\\ H_{\phi m}&=h_{\phi m}(r)\cos(n0)\sin(\omega t+ \zeta x),\\ H_{xm}&=h_{xm}(r)\sin(n0)\cos(\omega t+\zeta x). \end{split}  \tag{8}\]  
A multishell with a periodic alternation of two materials is further assumed (see Figure 1, lower panel) with \(r_{0}=100\) nm. All odd shells (\(m=2k+1\)) consist of one and the same material with elastic properties \(\lambda_{1}\), \(\mu_{1}\), density \(\rho_{1}\) and have the same thickness: \(\Delta r_{1}\). All even shells (\(m=2k\)) consist of the same material with elastic properties \(\lambda_{2}\), \(\mu_{2}\), density \(\rho_{2}\) and have the same thickness: \(\Delta r_{2}\). They are represented in Table 1. In what follows, the number of layers is denoted by \(N_{\rm L}\). The further procedure is a generalization of that in [65] for a two-shell tube onto an arbitrary number of shells. Substituting the eigenwaves of Equation (8) into the differential equations of motion (7), we exclude three coordinates (\(t\), \(x\), and \(\varphi\)) and arrive at a set of differential equations with respect to the radial coordinate \(r\) for the amplitude functions \(f_{m}(r)\) and \(h_{xm}(r)\) at \(m=1\), \(2\),..., \(N_{\rm L}\): \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parity \(i\) of the layer number** & **1** & **2** \\ \(m\) & & \\ \hline Material & InAs & GaAs \\ \(\lambda_{i}\), dyn/cm\({}^{2}\) & \(4.54\times 10^{11}\) & \(5.34\times 10^{11}\) \\ \(\mu_{i}\), dyn/cm\({}^{2}\) & \(1.90\times 10^{11}\) & \(3.285\times 10^{11}\) \\ \(\rho_{i}\), g/cm\({}^{3}\) & 5.68 & 5.317 \\ \(\Delta r_{i}\), nm & 5 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric and materials parameters of the multishell. \begin{table}
\begin{tabular}{c c} \hline \hline
**Characteristic of the layer** & **Denotation** \\
**with parity \(i\) of the number \(m\)** & **\(\rho_{1}^{2}=\rho^{2}/{\
u_{2}}^{2}-\zeta^{2}\)** \\ \hline Dispersion of the Rayleigh & \(\omega=v_{1}\zeta\) \\ waves \(\omega_{i}^{2}=0\) & \(\omega=v_{1}\zeta\) \\ \hline Dispersion of the Rayleigh & \(\omega=v_{2}\zeta\) \\ waves \(\beta_{i}^{2}=0\) & \(\omega=v_{2}\zeta\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Definitions of the wave characteristics in the materials. \[ \begin{split}\frac{\partial^{2}f_{m}}{\partial r^{2}}+\frac{1}{r} \frac{\partial f_{m}}{\partial r}+\left(\alpha_{m}^{2}-\frac{n^{2}}{r^{2}} \right)f_{m}=0,\\ \frac{\partial^{2}h_{mn}}{\partial r^{2}}+\frac{1}{r}\frac{ \partial h_{mn}}{\partial r}+\left(\beta_{m}^{2}-\frac{n^{2}}{r^{2}}\right)h_ {mn}=0\end{split}  \tag{9}\]  
and two similar equations for the amplitude functions \(h_{mn}(r)\) and \(h_{\varphi m}(r)\). Due to gauge invariance of the vector potential, one of the functions \(h_{m}(r)\), \(h_{\varphi m}(r)\), \(h_{\omega m}(r)\) can be selected arbitrarily; we use, like in [65], the calibration \(h_{rm}(r)=-h_{\varphi m}(r)\equiv h_{1m}(r)\). The differential Equations (9) are Bessel equations (see Chapter 9 in [67]). Their solutions can be represented in the general form:  
\[ \begin{split}& f_{m}(r)=A_{lm}Z_{n}\left(\left|\alpha_{m}\right|r \right)+B_{lm}W_{n}(\left|\alpha_{m}\right|r),\\ & h_{lm}(r)=A_{lm}Z_{n+1}(\left|\beta_{m}\right|r)+B_{2m}W_{n+1} (\left|\beta_{m}\right|r),\\ & h_{mn}(r)=A_{lm}Z_{n}\left(\left|\beta_{m}\right|r)+B_{3m}W_{n }(\left|\beta_{m}\right|r),\end{split}  \tag{10}\]  
where \(Z_{n}\) (\(W_{n}\)) are the Bessel functions of the first kind \(J_{n}\) (second kind \(Y_{n}\)) (p. 358 of [67]) for the real radial wave numbers \(\alpha_{m}\) and \(\beta_{m}\) and the modified Bessel functions \(I_{n}\) (\(K_{n}\)) (p. 374 of [67]) for the imaginary wave numbers \(\alpha_{m}\) and \(\beta_{m}\).