  A heuristic method to find asymptotic solutions to a system of non-linear
wave equations near null infinity is proposed. The non-linearities in this
model, dubbed good-bad-ugly, are known to mimic the ones present in the
Einstein field equations (EFE) and we expect to be able to exploit this method
to derive an asymptotic expansion for the metric in General Relativity (GR)
close to null infinity that goes beyond first order as performed by Lindblad
and Rodnianski for the leading asymptotics. For the good-bad-ugly model, we
derive formal expansions in which terms proportional to the logarithm of the
radial coordinate appear at every order in the bad field, from the second order
onward in the ugly field but never in the good field. The model is generalized
to wave operators built from an asymptotically flat metric and it is shown that
it admits polyhomogeneous asymptotic solutions. Finally we define stratified
null forms, a generalization of standard null forms, which capture the behavior
of different types of field, and demonstrate that the addition of such terms to
the original system bears no qualitative influence on the type of asymptotic
solutions found.
