On the momentum of solitons and vortex rings in a superfluid

TL;DR

This paper introduces a new renormalization method for calculating the momentum of solitons and vortex rings in superfluids, addressing convergence issues and comparing with existing approaches.

Contribution

It proposes a novel renormalization technique that explicitly separates vortex line contributions, improving momentum calculations for large vortex rings.

Findings

The method effectively handles convergence issues in momentum integrals.

Explicit vortex line term dominates for large rings.

Comparison with Jones and Roberts method shows consistency.

Abstract

This paper is devoted to the calculation of the momentum of localized excitations, such as solitons and vortex rings, moving in a superfluid. The direct calculation of the momentum by integration of the mass flux density results in a badly-converging integral. I suggest a method for the renormalization of the integral with the explicit separation of a term related to the vortex line. This term can be calculated explicitly and gives the main contribution for the rings whose size is large compared to the healing length. I compare my method with the Jones and Roberts prescription for the renormalization. I investigate the case of a uniform superfluid, and that of a superfluid in a cylindrical trap. I discuss the calculation of the jump in the phase of the order parameter and obtain a simple estimate for this jump for a large ring in the trap.