Nonlinear Klein-Gordon equation (${\partial }_t^2\varphi -{\partial }_x^2\varphi =f(\varphi )$)

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.

For $f(w)=-aw+b{w}^n$, with $a,b>0$ and integer $n>1$, a positive localized static pulse has $C^{n-1}=a(n+1)/(2b)$ and $B=(n-1)\sqrt{a}/2$. Writing the same profile as $[A{\cosh }^2(Bx){]}^{1/(1-n)}$ instead gives $A=2b/[a(n+1)]$. These are different amplitude conventions for the same solution. A Lorentz boost moves the pulse at speed $\tanh \beta$, preserves its peak height and contracts its width by $1/\cosh \beta$. Existence alone does not imply dynamical stability.