Hamilton's equations of motion of a vortex filament in the rotating Bose-Einstein condensate and their "soliton" solutions

TL;DR

This paper generalizes the equations of motion for vortex filaments in rotating Bose-Einstein condensates to arbitrary anharmonic traps, finding stationary and solitary wave solutions and analyzing their dependence on trap profiles.

Contribution

It introduces a variational form of the vortex filament equations for complex trap geometries and derives explicit solutions for stationary and moving vortex configurations.

Findings

Derived generalized equations of motion for vortex filaments in anharmonic traps.

Found explicit solutions for stationary vortices and solitary waves.

Analyzed how solutions depend on the condensate density profile.

Abstract

The equation of motion of a quantized vortex filament in a trapped Bose-Einstein condensate [A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A {\bf 62}, 063617 (2000)] has been generalized to the case of an arbitrary anharmonic anisotropic rotating trap and presented in a variational form. For condensate density profiles of the form $\rho=f(x^2+y^2+\mbox{Re\,}\Psi(x+iy))$ in the presence of the plane of symmetry $y=0$, the solutions $x(z)$ describing stationary vortices of U and S types coming to the surface and solitary waves have been found in quadratures. Analogous three-dimensional configurations of the vortex filament uniformly moving along the $z$ axis have also been found in strictly cylindrical geometry. The dependence of solutions on the form of the function $f(q)$ has been analyzed.