Domain wall evolution at nanowires in terms of 3D LLG equation initial-boundary problem

A theory of a domain wall creation and propagation is built on a linearized version of the transformed Landau-Lifshitz-Gilbert equation. The Lakshmanan-Nakamura stereo-graphic transform, after extra exponential transformation, and, next - linerization partially save information of the original nonlinearity that allows one to keep the domain wall dynamics, form and properties. For cylindrical-symmetric wire geometry, the conventional orthonormal Bessel basis, combined with projecting operators technique applied to subspaces of directed propagation of domain walls is constructed. The physically significant problems of the dynamics switching at points far and close from a wire ends are formulated and its solutions are presented in the frame of the Fourier method. Stationary solutions are found and the wall structure along the wire and propagation plots are drawn.