Multi-vortex traveling waves for the Gross-Pitaevskii equation and the Adler-Moser polynomials

For $N\le 34,$ we construct traveling waves with small speed for the Gross-Pitaevskii equation, by gluing $N(N+1)\mathrm{/}2$ pairs of degree $\pm 1$ vortices of the Ginzburg-Landau equation. The location of these vortices is symmetric in the plane and determined by the Adler-Moser polynomials, which has its origin in the study of Calogero-Moser system and rational solutions of the KdV equation. The construction still works for $N>34$, under the additional assumption that the corresponding Adler-Moser polynomial has no repeated root. It is expected that this assumption holds for any $N\in \mathbb{N}$.