Shape and dynamics of nonrelativistic vortex strings in parity-breaking media

The shape and dynamics of the nonrelativistic gauge vortex string in the parity-broken media is considered, upon reducing the problem to finding the extremum of the Abelian Higgs model effective action with the fixed B-type helicity of the gauge field. It is shown that in contrast with the case of the fixed A-type helicity, the static solution of the Ginzburg-Landau energy functional in the London limit is the helix with the specific relation between the curvature and torsion of the vortex line depending on the strength of the space parity violating contribution of the Lifshitz invariant. A nonlinear dynamical equation is linearized in case of small oscillations around the helical contour, and the polarization and dispersion law of the propagated waves are obtained.