  We construct an approximate solution of field equations in the Abelian Higgs
model which describes motion of a curved vortex. The solution is found to the
first order in the inverse mass of the Higgs field with the help of the
Hilbert-Chapman-Enskog method. Consistency conditions for the approximate
solution are obtained with the help of a classical Ward identity. We find that
the Higgs field of the curved vortex of the topological charge $n \geq 2$ in
general does not have single n-th order zero. There are two zeros: one is of
the (n-1)-th order and it follows a Nambu-Goto type trajectory, the other one
is of the first order and its trajectory in general is not of the Nambu-Goto
type. For $|n|=1$ the single zero in general does not lie on Nambu-Goto type
trajectory.
