Multivortex Traveling Waves for the Gross--Pitaevskii Equation and the Adler--Moser Polynomials

For each positive integer n≤34, we construct traveling waves with small speed for the Gross--Pitaevskii equation, by gluing $n(n+1)/2$ pairs of degree ±1 vortice of the Ginzburg--Landau equation. The location of these vortices is symmetric in the plane and determined by the roots of a special class of Adler--Moser polynomials, which are originated from the study of the 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 polynomials have no repeated roots. It is expected that this assumption holds for any n.