  We analyze a perturbation of the boundary Sine-Gordon model where two
boundary terms of different periodicities and scaling dimensions are coupled to
a Kondo-like spin degree of freedom. We show that, by pertinently engineering
the coupling with the spin degree of freedom, a competition between the two
boundary interactions may be induced, and that this gives rise to
nonpertubative phenomena, such as the emergence of novel quantum phases:
indeed, we demonstrate that the strongly coupled fixed point may become
unstable as a result of the "deconfinement" of a new set of phase-slip
operators -the short instantons- associated with the less relevant boundary
operator.
  We point out that a Josephson junction network with a pertinent impurity
located at its center provides a physical realization of this boundary double
Sine-Gordon model. For this Josephson junction network, we prove that the
competition between the two boundary interactions stabilizes a robust finite
coupling fixed point and, at a pertinent scale, allows for the onset of $4e$
superconductivity.
