On the momentum of solitons and vortex rings in a superfluid
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e Momentum of Solitons and Vortex Rings in a Superfluid1 L. P. Pitaevskii Kapitza Institute for Physical Problems Russian Academy of Sciences, Moscow, 119334 Russia Dipartimento di Fisica, Universitá di Trento and INOCNR BEC Center, I38123 Trento, Povo, Italy email: [email protected] Received May 28, 2014
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 badlyconverging 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 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. Contribution for the JETP special issue in honor of A.F. Andreev’s 75th birthday DOI: 10.1134/S1063776114120152 1
1. INTRODUCTION
of a fluid, i.e. using the equation:
One of the characteristic properties of a superfluid is the possible existence of localized stationary excita tions. These can be solitons, vortex rings, solitonic vortices, or other more complicated objects. The study of the vortex rings has resulted in important progress in the understanding of the physics of superfluid helium. A very interesting possibility arose after the creation of new superfluid systems like ultracold Bose and Fermi gases, which are confined in traps. The dynamics of the excitations in these systems is an important theo retical and experimental problem due to the presence of external fields. At the same time, due to the dilute ness of the gases, one can develop a microscopic the ory which is sufficiently detailed. If the trapping field changes slowly enough, it is convenient to solve the problem in two steps. As a first step, it is reasonable to find the energy and momentum for an excitation in a uniform fluid. (In the case of an elongated trap, the fluid is uniform in the axial direction.) Second, one can analyze the motion of the excitations using the semiclassical equations of motion. Depending on the nature of the problem, the energy should be expressed in terms of momentum or velocity. However, a difficulty arises which had already been encountered in classical hydrodynamics (see for instance [1, Section 11] or [2, Section 6]). If one tries to calculate the momentum by integration of the mass flux density j over the volume 1 The article is published in the original.
p =
∫ jd x, 3
(1)
then, for stationary motion, the integrand does not decrease fast enough and the value of the integral depends on the shape of the integration domain. Dif ferent prescriptions have been suggested to overcome this
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