In this article we prove that 2-soliton solutions of the sine-Gordon equation
(SG) are orbitally stable in the natural energy space of the problem. The
solutions that we study are the {\it 2-kink, kink-antikink and breather} of SG.
In order to prove this result, we will use B\"acklund transformations
implemented by the Implicit Function Theorem. These transformations will allow
us to reduce the stability of the three solutions to the case of the vacuum
solution, in the spirit of previous results by Alejo and the first author,
which was done for the case of the scalar modified Korteweg-de Vries equation.
However, we will see that SG presents several difficulties because of its
vector valued character. Our results improve those in Alejo et al., and give a
first rigorous proof of the stability in the energy space of SG 2-solitons.Comment: 55pp, 12 fig
