## 6 Conclusions and Future Challenges  
In the present work, we have revisited the partially \(\mathcal{PT}\)-symmetric setting originally proposed in [19] and have attempted to provide a systematic analysis of the existence, stability and evolutionary dynamics of the nonlinear modes that arise in the presence of such a potential for both self-focusing and self-defocusing nonlinearities. It was found that all three linear modes generate nonlinear counterparts. Generally, the defocusing case was found to be more robustly stable than the focusing one. In the former, two of the branches were stable for all the values of the frequency considered, while in the focusing case, all three branches developed instabilities sufficiently far from the linear limit (although all of them were spectrally stable close to it). The instabilities could be of different types, both oscillatory (as for branch 2) and exponential (as for branch 3) or even of mixed type (as for branch 1). The resulting oscillatorily or exponentially unstable dynamics, respectively, led to the gain overwhelming the dynamics and leading to indefinite growth in the one or two of the gain peaks of our four-peak potential.  
Naturally, there are numerous directions that merit additional investigation. For instance, and although this would be of less direct relevance in optics, partial \(\mathcal{PT}\) symmetry could be extended to three dimensions. There, it would be relevant to appreciate the differences between potentials that are partially \(\mathcal{PT}\) symmetric in one direction vs. those partially \(\mathcal{PT}\) symmetric in two directions. Another relevant case to explore in the context of the present mode is that where a \(\mathcal{PT}\) phase transition has already occurred through the collision of the second and third linear eigenmode considered herein. Exploring the nonlinear modes and the associated stability in that case would  
Figure 8: This figure is similar to Figure 5 (with an evolution up to distance \(z=300\)). Here, the plots correspond to the evolution of the branch 2 solution for the value \(\mu\approx 0.244\). In the top right plot, the colors follow the same pattern as in Figure 5 representing the four peaks. While the gain side of the solution seems to dominate, we also observe a quick increase in the spatial footprint of the solution. Shortly after the plotted time interval, the solution interferes with the boundary of the computed spatial grid.  
be an interesting task in its own right. Such studies are presently under consideration and will be reported in future publications.  
**Acknowledgments:** [PERSON] gratefully acknowledges the support of NSF-PHY-1602994, the Alexander von Humboldt Foundation, and the ERC under FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-605096). Both authors express their gratitude to Professor [PERSON] for initiating their interest in this direction and for numerous relevant discussions.  
**Author Contributions:** [PERSON] and [PERSON] designed the research; [PERSON] carried out the numerical computations; both authors contributed substantially to the explanation of the results and to writing of the work.  
**Conflicts of Interest:** The authors declare no conflict of interest.