  We study a semi-linear version of the Skyrme system due to Adkins and Nappi.
The objects in this system are maps from $(1+3)$-dimensional Minkowski space
into the $3$-sphere and 1-forms on $\mathbb{R}^{1+3}$, coupled via a Lagrangian
action. Under a co-rotational symmetry reduction we establish the existence,
uniqueness, and unconditional asymptotic stability of a family of stationary
solutions $Q_n$, indexed by the topological degree $n \in \mathbb{N} \cup
\{0\}$ of the underlying map. We also prove that an arbitrarily large
equivariant perturbation of $Q_n$ leads to a globally defined solution that
scatters to $Q_n$ in infinite time as long as the critical norm for the
solution remains bounded on the maximal interval of existence given by the
local Cauchy theory. We remark that the evolution equations are super-critical
with respect to the conserved energy.
