  We consider the Yang-Mills problem on $\mathbb{R}^{1+4}$ with gauge group
$SO(4)$. In an appropriate equivariant reduction, this Yang-Mills problem
reduces to a single scalar semilinear wave equation. This semilinear equation
admits a one-parameter family of solitons, each of which is a re-scaling of a
fixed solution. In this work, we construct a class of solutions, each of which
consists of a soliton whose length scale is asymptotically constant, coupled to
large radiation, plus corrections which slowly decay to zero in the energy
norm. Our class of solutions includes ones for which the radiation component is
only "logarithmically" better than energy class. As such, the solutions are not
constructed by apriori assuming the length scale to be constant. Instead, we
use an approach similar to a previous work of the author regarding wave maps.
In the setup of this work, the soliton length scale asymptoting to a constant
is a necessary condition for the radiation profile to have finite energy. An
interesting point of our construction is that, for each radiation profile,
there exist one-parameter families of solutions consisting of the radiation
profile coupled to a soliton, which has any asymptotic value of the length
scale.
