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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 {\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 $ρ=f(x^2+y^2+\mbox{Re\,}Ψ(x+iy))$ 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.