Generalized Adler-Moser Polynomials and Multiple vortex rings for the Gross-Pitaevskii equation

TL;DR

This paper constructs new finite energy traveling wave solutions with multiple vortex rings for the 3D Gross-Pitaevskii equation, using generalized Adler-Moser polynomials to describe vortex ring locations.

Contribution

It introduces a novel class of solutions with vortex rings characterized by roots of generalized Adler-Moser polynomials, extending classical polynomial frameworks.

Findings

Solutions feature 2n+1 vortex rings with specific orientations.

Vortex ring positions are determined by roots of generalized Adler-Moser polynomials.

The approach generalizes classical polynomial methods for vortex configurations.

Abstract

New finite energy traveling wave solutions with small speed are constructed for the three dimensional Gross-Pitaevskii equation \begin{equation*} i\Psi_t= \Delta \Psi+(1-|\Psi|^2)\Psi, \end{equation*} where $\Psi$ is a complex valued function defined on ${\mathbb R}^3\times{\mathbb R}$. These solutions have the shape of $2n+1$ vortex rings, far away from each other. Among these vortex rings, $n+1$ of them have positive orientation and the other $n$ of them have negative orientation. The location of these rings are described by the roots of a sequence of polynomials with rational coefficients. The polynomials found here can be regarded as a generalization of the classical Adler-Moser polynomials and can be expressed as the Wronskian of certain very special functions. The techniques used in the derivation of these polynomials should have independent interest.