  A domain wall of relative phase in a flattened harmonically-trapped
Bose-Einstein condensed mixture of two atomic hyperfine states, subject to a
stationary Rabi coupling of intensity $\Omega$, is predicted to decay through
two different mechanisms. For small values of $\Omega$ the instability has an
energetic nature and is associated with the formation of a vortex-antivortex
pair of the same atomic hyperfine states, whose motion inside the trap causes
the emergence of magnetization, the bending of the domain wall and its
consequent fragmentation. For large values of $\Omega$ the domain wall instead
undergoes a dynamic snake instability, caused by the negative value of its
effective mass and results in the fast fragmentation of the wall into smaller
domain walls confining vortex pairs of different atomic species. Numerical
predictions are given by solving the time-dependent Gross-Pitaevskii equation
in experimentally available configurations of mixtures of sodium atomic gases.
