Asymptotic stability of Bloch wall for dispersive Landau-Lifshitz equation

We consider the Bloch wall stationary solutions to the dispersive Landau-Lifshitz equation with Dzyaloshinskii-Moriya interaction (DMI). First, we establish a dichotomy for the orbital stability and instability depending on the value of a coupling constant. Then, under symmetric perturbations, we prove that the Bloch wall is asymptotically stable in the energetically favorable regime.