  We construct series solutions to all orders for breathers of Klein-Gordon
equations, in powers of an amplitude parameter epsilon, under a sign condition
on the coefficients of the expansion of the nonlinearity. All terms may be
computed thanks to the properties of a 1D Schr\"odinger equation with
two-soliton potential. We prove that this series is free of poles in the unit
disc of the complex epsilon plane if and only if the equation is essentially
equivalent to the sine-Gordon equation. This appears to be the only result that
explains both what makes the sine-Gordon equation special from a PDE point of
view, and at the same time, why approximate, long-lived breathers exist. In a
related paper (Classical and Quantum Gravity, 25 (2008) 245004 ) we have
extended this approach to address the soliton star problem in General
Relativity.
