Shape-preserving solutions for quantum vortex motion under localized induction approximation

Abstract

The motion of a quantum vortex in superfluid helium is considered in the localized induction approximation. In this approximation the instantaneous velocity of quantum vortex is proportional to the local curvature and is parallel to the vector, which is a linear combination of the local binormal and the principal normal to the vortex line. The motion in the direction of the principal normal is specific for a quantum vortex and implies that the vortex shrinks, in contrast to the classical vortex in an ideal fluid. In the present work we deal with two four-parameter classes of shape-preserving solutions (one with increasing and one with decreasing spatial scale) resulting from equations governing the curvature and the torsion. The solutions describe vortex lines whose motion is equivalent to a transformation being a superposition of a homothety and a rotation. In a particular case when the transformation is a pure homothety, we find analytic solutions for the curvature and the torsion. In the general case, when the transformation is a superposition of a nontrivial rotation and a homothety, the asymptotics of the solutions of the first class are given explicitly and are related to the parameters characterizing the transformation. It is found that the solutions of the second class (with decreasing scale) either have asymptotes or are periodic (when the transformation is a pure homothety) or else exhibit chaotic behavior.