Figure 1
From: 1D solitons in cubic-quintic fractional nonlinear Schrödinger model
The variational solution (21) for the fractional soliton for \(\omega =\omega _0\) (\(\lambda =1\), left panel) and \(\omega =5\omega _0\) (\(\lambda =1/\sqrt{5}\), right panel), plotted for Lévy indices \(\alpha\) (legend in the left panel) down to their critical values \(\alpha _{cr}(\lambda )\) for soliton existence. Namely, for \(\omega =\omega _0\) (\(\lambda =1\)) \(\alpha _{cr} \approx 0.6266\), while for \(\omega =5\omega _0\) (\(\lambda =1/\sqrt{5}\)) \(\alpha _{cr} \approx 0.6491\). We plot the critical soliton textures for \(\alpha\) (shown near the corresponding curves), slightly larger that the critical values, for better visualization. Curves, labeled ”unstable”, correspond to Lévy indices, for which \(dN/d\omega >0\), i.e. VK stability criterion does not fulfilled.
