# Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses
## 4 Characteristics of Solitons
### Multi-Peak Solitons  
The Hermite-Gaussian soliton solutions with multipeaks in fact have been discussed in nonlocally nonlinear media with positively-definitiing attenuating response functions [44; 45]. However, the upper thresholds of the peak number of the stable solitons in nonlocally nonlinear media are different, which closely depend on the response function of media. For instance, in the nonlocal system with the Gaussian response, the multipeak solitons with any peak number [4] are stable [44]. In nematic liquid crystals with positive dielectric anisotropy, the response function is an exponential decay one for the \((1+1)\) case, and the peak-number of the solitons supported is only less than five [44]. In the nonlocally nonlinear system with oscillatory responses described by Equations (1) and (2), we found the upper thresholds of the peak number of the stable solitons are five and four in the cases of \(s=-1\) and \(1\), respectively [30].  
The variational approach can be applied to find solitons and discuss their stabilities. The Hermite-Gaussian solitons of the \((1+1)\)-dimensional NNLSE (7) are of the following profiles:  
\[u_{n}=\Bigg{(}\frac{P_{nc}\sqrt{2n+1}}{2^{n}n!\sqrt{\pi}}\Bigg{)}^{1/2}H_{n}( \sqrt{2n+1}x)\exp\biggl{[}-\frac{(2n+1)x^{2}}{2}\biggr{]}. \tag{27}\]  
The critical power is obtained as follows:  
\[P_{nc}=\frac{(2n+1)(\sqrt{\pi}2^{n}n!)^{2}}{sN^{(n)}(1,\sigma_{0})}, \tag{28}\]  
where function \(N^{(n)}\) is the \(N_{n}\) given in Ref. [30]. The degree of nonlocality range within which HG solitons exist is obtained by the variational approach, which is summarized in Table 1, where the numerical results are also given for a comparison.  
Multipeak solitons can be numerically obtained for two cases of \(s=\pm 1\), which are shown in Figures 12 and 13, respectively. Clearly, the multipeak solitons in the case of \(s=-1\) exhibit Hermite-Gaussian structures and are in good agreement with numerical ones. While, the profiles of solitons in the case of \(s=1\) deviate from the Hermite-Gaussian form, especially when \(\sigma_{0}\) becomes large. Likely, for multipeak solitons, the propagation constants and the slope of the power-propagation constant are both negative, which is shown in Figure 14. By linear stability analyses, the ranges of the degree of nonlocality \(\sigma_{0}\) within which the stable multi-peak solitons exist are summarized in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & & \(\mathbf{n=0}\) & \(\mathbf{n=1}\) & \(\mathbf{n=2}\) & \(\mathbf{n\geq 3}\) \\ \hline \(s=-1\) & variational & \((1.05,+\infty)\) & \((1.06,+\infty)\) & \((1.06,+\infty)\) & \((1.06,+\infty)\) & \((1.06,+\infty)\) \\  & numerical & \((1.05,+\infty)\) & \((1.05,+\infty)\) & \((1.05,+\infty)\) & \((1.05,+\infty)\) \\ \hline \(s=+1\) & variational & \((0,0.77)\) & \((0.38,0.79)\) & \((0.39,0.79)\) & \((0.39,0.79)\) \\  & numerical & \((0.05,0.78)\) & \((0.38,0.78)\) & \((0.39,0.78)\) & \((0.40,0.78)\) \\ \hline \end{tabular} Note: self-focusing nonlinearity is exhibited when \(\sigma_{0}\in(\sigma_{c},+\infty)\) for \(s=-1\), and \([0,\sigma_{c}]\) for \(s=1\) (for example, \(\sigma_{c}|_{s-1}=0.82\)).  
\end{table}
Table 1: The ranges of \(\sigma_{0}\) within which the HG-type solitons exist.  
Figure 12: Profiles of the numerical multi–peak solitons \(u(x)\) (the solid red curves) for \(s=-1\) and \(w_{m}=5\). (**a–d**) for \(n=1,4,5,6\) and \(\sigma_{0}=1.13,1.43,3.52,1.08\), respectively. The variational results (the dashed black curves) with the same parameters are also given for comparison. Soliton in (**b**) is stable, while the ones in the other three figures are unstable.  
Figure 13: Profiles of the numerical multi–peak solitons \(u(x)\) (the solid red curves) for \(s=1\) and \(w_{M}=10\). (**a–d**) for \(n=2,3,5,6\) and \(\sigma_{0}=0.74,0.48,0.75,0.51\), respectively. The corresponding numerical results are denoted in the same manner as in Figure 5. Soliton in (**b**) is stable, while the ones in the other three figures are unstable.