= Sine-Gordon kink
{c}
{title2=$\phi_K=4\arctan e^{x-a}$}
The static <kink> $\phi_K(x)=4\arctan e^{x-a}$ joins adjacent <scalar-field vacua> $0$ and $2\pi$. It obeys $\phi_K'=2\operatorname{sech}(x-a)$ and has <topological charge> $Q=[\phi(+\infty)-\phi(-\infty)]/(2\pi)=1$. In the <Sine-Gordon theory> normalization with physical mass scale $m$ and coupling $\beta$, its classical <mass> is $8m/\beta^2$. A <Lorentz boost> gives $\phi_K=4\arctan\exp[(x-vt-a)/\sqrt{1-v^2}]$. Spatial reflection gives an <antikink>.
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