Klein–Gordon equation

Summary

The Klein–Gordon equation $u_{tt} - \Delta u + m^2 u = 0$ is the wave equation with a mass term. It is the relativistic wave equation of a free scalar particle of mass $m$, written in units with $c = \hbar = 1$. It is hyperbolic, so signals travel no faster than light, and for $m > 0$ it is also dispersive: waves of different frequencies travel at different speeds. Its nonlinear versions, such as $u_{tt} - \Delta u + m^2 u = \pm \abs{u}^{p-1}u$, are model problems in nonlinear wave theory.