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.
Categories: math.AP