Hollow Vortices as Nonlinear Waves

In this chapter, we discuss recent progress on the existence of steady collections of hollow vortices made by the author with Chen et al. (Desingularization and global continuation for hollow vortices, Preprint, arXiv:2303.03570, 2023). In doing so, we endeavor to highlight the mathematical similarities between the hollow vortex system on the one hand and traveling water waves beneath vacuum on the other. While hollow vortices are not nonlinear waves in the strictest sense, many of the strategies developed in the study of water waves are extremely helpful in their analysis. Several extensions of the results in Chen et al. (Desingularization and global continuation for hollow vortices, Preprint, arXiv:2303.03570, 2023) are also provided, and we outline some important open problems to be considered in future work.