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In this work we construct the wobble exact solution of sine-Gordon equation by means of Bäcklund Transformations. We find the parameters of the transformations corresponding to the Bianchi diagram for the wobble as a particular 3-soliton solutions. We show that this solution agrees with the wobbles obtained by Kälbermann and Segur by means of the Inverse Scattering Transform, and by Ferreira et al. using the Hirota method. The new formulation introduced allows to identify easily the parameters that define the building blocks of this solution–a kink and a breather, and can be used in further studies of this solution in the perturbed sine-Gordon equation.

1. Introduction. Klein-Gordon type models bear topological solutions called kink (or antikink) solitons, which are solitary waves that connect two minima of the potential. These solutions evolve in time with constant velocity without changing their shape. There are also oscillatory, non-topological solutions called breathers, which are bound states of a kink and an antikink. Yet another solution is the “wobble”, which can be interpreted as a nonlinear superposition of a kink and a breather [36, 5, 18].