Klein–Gordon equation

Remarks

Example: for $k \in \R^n$ and $\omega = \sqrt{\abs{k}^2 + m^2}$, the plane wave $u(x, t) = e^{i(k \cdot x - \omega t)}$ is a solution, since $u_{tt} = -\omega^2 u$ and $\Delta u = -\abs{k}^2 u$, so $u_{tt} - \Delta u + m^2 u = (-\omega^2 + \abs{k}^2 + m^2) u = 0$. Non-example: the plane wave $e^{i(k \cdot x - \abs{k} t)}$ solves the wave equation but not the Klein–Gordon equation with $m > 0$, since then the left side is $m^2 u \ne 0$. For $m = 0$ the equation is the wave equation with speed $1$. The conserved energy is $E(t) = \frac{1}{2}\int_{\R^n} \big(\abs{u_t}^2 + \abs{\nabla u}^2 + m^2 \abs{u}^2\big)\, dx$.