  The boundary correlation functions for a Quantum Field Theory (QFT) in an
Anti-de Sitter (AdS) background can stay conformally covariant even if the bulk
theory undergoes a renormalization group (RG) flow. Studying such correlation
functions with the numerical conformal bootstrap leads to non-perturbative
constraints that must hold along the entire flow. In this paper we carry out
this analysis for the sine-Gordon RG flows in AdS$_2$, which start with a free
(compact) scalar in the UV and end with well-known massive integrable theories
that saturate many S-matrix bootstrap bounds. We numerically analyze the
correlation functions of both breathers and kinks and provide a detailed
comparison with perturbation theory near the UV fixed point. Our bounds are
often saturated to one or two orders in perturbation theory, as well as in the
flat-space limit, but not necessarily in between.
