  We consider an infinite ferromagnetic nanowire, with an energy functional $E$
with easy-axis in the direction $e_1$ and a constant external magnetic field
$E_{ext} = h_0 e_1$ along the same direction. The evolution of its
magnetization is governed by the Landau-Lifshitz-Gilbert equation (LLG)
associated to $E$. Under some assumptions on $h_0$, we prove the existence of
stationary solutions with the same limits at infinity, their uniqueness up to
the invariances of the equation and the instability of their orbits with
respect to the flow. This property gives interesting new insights of the
behavior of the solutions of (LLG), which are completed by some numerical
simulations and discussed afterwards, in particular regarding the stability of
2-domain wall structures proven in [4] and more generally the interactions
between domain walls.
