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We develop a perturbation theory that describes bound states of solitonslocalized in a confined area. External forces and influence of inhomogeneitiesare taken into account as perturbations to exact solutions of the sine-Gordonequation. We have investigated two special cases of fluxon trapped by amicroresistor and decay of a breather under dissipation. Also, we have carriedout numerical simulations with dissipative sine-Gordon equation and madecomparison with the McLaughlin-Scott theory. Significant distinction betweenthe McLaughlin-Scott calculation for a breather decay and our numerical resultindicates that the history dependence of the breather evolution can not beneglected even for small damping parameter

Perturbation theory for localized solutions of the sine-Gordon equation: Decay of a breather and pinning by a microresistor