Uniqueness of one-dimensional N\'eel wall profiles

We study the domain wall structure in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Using the reduced one-dimensional thin film micromagnetic model, we analyze the critical points of the obtained non-local variational problem. We prove that the minimizer of the one-dimensional energy functional in the form of the N\'eel wall is the unique (up to translations) critical point of the energy among all monotone profiles with the same limiting behavior at infinity. Thus, we establish uniqueness of the one-dimensional monotone N\'eel wall profile in the considered setting. We also obtain some uniform estimates for general one-dimensional domain wall profiles.