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In this paper we prove that the static solution of the Cauchy problem for amassless real scalar field that is sourced by a point charge in $1+1$dimensions is asymptotically stable under perturbation by compactly-supportedincoming radiation. This behavior is due to the process of back-reaction.Taking the approach of Kiessling, we rigorously derive the expression for theforce on the particle from the principle of total energy-momentum conservation.We provide a simple, closed form for the self-force resulting fromback-reaction, and show that it is restorative, i.e. proportional to negativevelocity, and causes the charge to return to rest after the radiation passesthrough. We establish these results by studying the joint evolution problem forthe particle-scalar field system, and proving its global well-posedness and theclaimed asymptotic behavior.

Joint evolution of a Lorentz-covariant massless scalar field and its point-charge source in one space dimension