The Maxwell-scalar field system near spatial infinity

TL;DR

This paper investigates how nonlinear interactions in the Maxwell-scalar field system affect the regularity and asymptotic behavior of solutions near spatial infinity, revealing new singularities and logarithmic terms at null infinity.

Contribution

It provides a detailed analysis of the nonlinear effects on asymptotic expansions near spatial infinity, highlighting the emergence of singularities and new logarithmic terms.

Findings

Nonlinear interactions increase singularities at the conformal boundary.

Discovery of new logarithmic terms in asymptotic expansions.

Implications for the peeling behavior of fields at null infinity.

Abstract

We make use of Friedrich's representation of spatial infinity to study asymptotic expansions of the Maxwell-scalar field system near spatial infinity. The main objective of this analysis is to understand the effects of the non-linearities of this system on the regularity of solutions and polyhomogeneous expansions at null infinity and, in particular, at the critical sets where null infinity touches spatial infinity. The main outcome from our analysis is that the nonlinear interaction makes both fields more singular at the conformal boundary than what is seen when the fields are non-interacting. In particular, we find a whole new class of logarithmic terms in the asymptotic expansions which depend on the coupling constant between the Maxwell and scalar fields. We analyse the implications of these results on the peeling (or rather lack thereof) of the fields at null infinity.