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

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 62, 063617 (2000)] has been generalized to the case of an arbitrary anharmonic anisotropic rotating trap and presented in the variational form. For condensate density profiles of the form $\rho=f(x^2+y^2+\mathrm{Re}\,\Psi(x+iy))$ in the presence of the plane of symmetry $y=0$, the solutions $x(z)$ describing stationary vortices of $\mathrm{U}$ and $\mathrm{S}$ 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.