  We propose a simple model consisting of a magnetic domain wall
proximity-coupled to an $s$-wave superconductor for realization of Majorana
zero-energy modes. A spin-dependent gauge transformation translates the
rotating magnetic profile through the domain wall to effective spin-orbit and
Zeeman terms. The Hamiltonian breaks time reversal and chiral symmetries, while
preserving particle-hole symmetry, placing itself into topological D class
characterized by the $\mathbb{Z}_{2}$ topological invariant for quasi
one-dimensional system. The low-energy sector of the model maps to the one
isomorphic with Kitaev Hamiltonian. The existence and localization of Majorana
zero modes in the nontrivial phase are demonstrated explicitly and we obtain
the topological phase diagram with extended regime of nontrivial phase and
surprising occurrence of a re-entrance phase transition. Our calculation shows
that the system can be easily tuned between trivial and topological ground
states and can be implemented experimentally to realize non-Abelian statistics.
