Stability and instability of solitary-wave solutions for the nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon (KG) equation,${\partial }_{tt}u-\Delta u+u=|u{|}^{p-1}u,(t,x)\in R\times R^d$ is shown in the present paper to possess the solitary-wave solutions in the form of $e^{i\omega t}{\varphi }_{\omega ,\overrightarrow{c}}(x-\overrightarrow{c}t)$ with the parameters ω and $\overrightarrow{c}\in R^d$ satisfying $\left|\omega \right|<\sqrt{1-|\overrightarrow{c}{|}^2}$ and $\left|\overrightarrow{c}\right|<1$. By employing a new localized virial identity combined with the coercivity and modulation argument, it is demonstrated here that there exists a critical frequency ${\omega }_{⁎}\left(\right|\overrightarrow{c}\left|\right)$ such that these localized solitary waves, when considered as solutions of the initial-value problem for the nonlinear KG equation, is dynamically stable when $1<p<1+\frac{4}{d},\left|\omega \right|>{\omega }_{⁎}\left(\right|\overrightarrow{c}\left|\right)$ and is dynamically unstable when $1<p<1+\frac{4}{d},0<\left|\omega \right|\le {\omega }_{⁎}\left(\right|\overrightarrow{c}\left|\right)$ or $1+\frac{4}{d}\le p<1+\frac{4}{d-2}$ ($d\ge 3$ or $1<p<\infty$ if $d=1,2$) to small perturbations.