High order asymptotic expansions of a good–bad–ugly wave equation

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