  We consider a spatially inhomogeneous sine-Gordon equation with a
time-periodic drive, modeling a microwave driven long Josephson junction with
phase-shifts. Under appropriate conditions, Josephson junctions with
phase-shifts can have a spatially nonuniform ground state. In recent reports,
it is experimentally shown that a microwave drive can be used to measure the
eigenfrequency of a junction's ground state. Such a microwave spectroscopy is
based on the observation that when the frequency of the applied microwave is in
the vicinity of the natural frequency of the ground state, the junction can
switch to a resistive state, characterized by a non-zero junction voltage. It
was conjectured that the process is analogous to the resonant phenomenon in a
simple pendulum motion driven by a time periodic external force. In the case of
long junctions with phase-shifts, it would be a resonance between the internal
breathing mode of the ground state and the microwave field. Nonetheless, it was
also reported that the microwave power needed to switch the junction into a
resistive state depends on the magnitude of the eigenfrequency to be measured.
Using multiple scale expansions, we show here that an infinitely long Josephson
junction with phase-shifts cannot be switched to a resistive state by microwave
field with frequency close to the system's eigenfrequency, provided that the
applied microwave amplitude is small enough, which confirms the experimental
observations. It is because higher harmonics with frequencies in the continuous
spectrum are excited, in the form of continuous wave radiation. The presence of
applied microwaves balances the nonlinear damping, creating a stable breather
mode oscillation. We confirm our analytical results numerically.
