We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity${\partial }_t^{2}u-{\partial }_x^{2}u+u-|u{|}^{p-1}u+|u{|}^{q-1}u=0,\text{on}[0,\infty )\times \mathbb{R},$ where $1<q<p<\infty$. 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 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.

我们考虑具有双倍聚焦-散焦非线性的一维非线性 Klein-Gordon 方程${\partial }_{吨}^2你-{\partial }_X^2你+你-|你{|}^{磷-1}你+|你{|}^{q-1}你=0,\text{在}[0,\infty )\times \mathbb{电阻},$ 在哪里 $1<q<磷<\infty$. 主要结果表明能量空间的稳定性$H^1(\mathbb{电阻})\times {升}^2(\mathbb{电阻})$具有不同速度的去耦孤立波的总和，直到自然不稳定性。该证明的灵感来自于 Martel、Merle 和 Tsai [14]、[15] 在类似上下文中为广义 Korteweg-de Vries 方程和非线性薛定谔方程开发的技术。然而，这种策略对波型方程的适应需要引入适应洛伦兹变换的新能量函数。