  The dynamics of interacting quantized vortex filaments in a rotating trapped
Bose-Einstein condensate, which is in the Thomas-Fermi regime at zero
temperature and described by the Gross-Pitaevskii equation, is considered in
the hydrodynamic "anelastic" approximation. In the presence of a smoothly
inhomogeneous array of filaments (vortex lattice), a non-canonical Hamiltonian
equation of motion is derived for the macroscopically averaged vorticity, with
taking into account the spatial non-uniformity of the equilibrium condensate
density determined by the trap potential. A minimum of the corresponding
Hamiltonian describes a static configuration of deformed vortex lattice against
a given density background. The minimum condition is reduced to a vector
nonlinear partial differential equation of the second order, for which some
approximate and exact solutions are found. It is shown that if the condensate
density has an anisotropic Gaussian profile then equation of motion for the
averaged vorticity admits solutions in the form of a spatially uniform vector
with a nontrivial time dependence. An integral representation is obtained for
the matrix Green function determining the non-local Hamiltonian of a system of
arbitrary shaped vortex filaments in a condensate with Gaussian density. ... A
simple approximate expression for the two-dimensional Green function is
suggested at rather arbitrary density profile, and its successful comparison to
the exact result in the Gaussian case is done. Approximate equations of motion
are derived which describe a long-wave dynamics of interacting vortex filaments
in condensates with the density depending on the transverse coordinates only.
