 Nonlinear Klein-Gordon equation

A local scalar nonlinearity preserves Lorentz covariance of the wave operator when the wave speed is one. If $f=F^{\prime }$, the equation follows from $\mathcal{L}=({\partial }_t\varphi {)}^2/2-({\partial }_x\varphi {)}^2/2+F(\varphi )$. The physical potential is $-F$, so its sign must be distinguished from the sign written in the Lagrangian.