Multivortex traveling waves for the Schrödinger map equation

Abstract

We construct traveling wave solutions for the Schrödinger map equation in R2. These solutions have n(n + 1)/2 pairs of degree ±1 vortices. The locations of those vortices are symmetric in the plane and determined by the roots of a special class of Adler–Moser polynomials. With a few modifications, a similar construction allows for the creation of traveling wave solutions of the Schrödinger map equation in R3. These solutions have the shape of 2n + 1 vortex rings, whose locations are given by a sequence of polynomials with rational coefficients and are far away from each other.