  We consider the one-dimensional nonlinear Klein-Gordon equation with a double
power focusing-defocusing nonlinearity
  \begin{equation*}
  \partial_{t}^{2}u-\partial_{x}^{2}u+u-|u|^{p-1}u+|u|^{q-1}u=0,\quad
\mbox{on}\ [0,\infty)\times \mathbb{R},
  \end{equation*}
  with $1<q<p<\infty$. The main result states the stability in the energy space
$H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$ of the sums of decoupled solitary
waves with different speeds, up to the natural instabilities. The proof is
inspired by the techniques developed for the generalized Korteweg-de Vries
equation and the nonlinear Schr\"odinger equation in a similar context by
Martel, Merle and Tsai [14,15]. However, the adaptation of this strategy to a
wave-type equation requires the introduction of a new energy functional adapted
to the Lorentz transform.
