= Solution
The <Lagrangian density> is <Lorentz invariant>, so a scalar-valued static solution can be composed with a <Lorentz transformation>. Let $X=\gamma(x-vt-x_0)$, with $\gamma=(1-v^2)^{-1/2}$ and $|v|<1$. If $\phi(x,t)=f(X)$, the <chain rule> gives
$$
(\partial_t^2-\partial_x^2)f(X)=\gamma^2(v^2-1)f''(X)=-f''(X).
$$
The boosted <equation of motion> is therefore exactly the static one already verified in <5/b/solution>, and
$$
\boxed{\phi(x,t)=\frac4b\arctan\exp\left[k\gamma(x-vt-x_0)\right]}
$$
is a solution. The center moves as $x=x_0+vt$: $v$ is the physical <velocity> in units where the limiting speed is one. The <kink> width contracts from $k^{-1}$ to $(\gamma k)^{-1}$, while its <topological charge> remains unchanged.
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