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

High Order Asymptotic Expansions of a Good-Bad-Ugly Wave Equation