The perturbed sine-Gordon breather equation integrated by Riemann's method

For the perturbed nonlinear sine-Gordon equation a general first-order perturbation solution in the neighbourhood of the unperturbed breather solution is given. The inhomogeneous linear hyperbolic differential equation is solved by Riemann's method. For the determination of the Riemann function no methods of inverse scattering theory are used. Instead, the Bäcklund transformation and a novel inversion relation are applied. The Riemann function may be expressed in terms of Lommel functions of two variables. It is shown that the so formulated Riemann function has, unlike the discrete part, the correct symmetry. As an example, the asymptotic solution for a low-amplitude breather under a constant perturbation is given, showing that plane waves are radiated to both sides of the breather.