Conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity

We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity${\partial }_t^{2}u-{\partial }_x^{2}u+u-|u{|}^{p-1}u+|u{|}^{q-1}u=0,\text{on}[0,\infty )\times R,$ where $1<q<p<\infty$. The main result states the stability in the energy space $H^1\left(R\right)\times L^2\left(R\right)$ 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 Schrödinger 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.