= Sine-Gordon threshold kink-antikink solution
{c}
{title2=$\phi=4\arctan[(t-t_0)/\cosh x]$}
The zero-relative-speed limit of the <Sine-Gordon kink-antikink scattering solution> is a <separatrix> at twice the rest-kink energy. A <Sine-Gordon Bäcklund transformation> of the static <kink> with equal parameter gives $p_x=-p\tanh x$, $p_t=\operatorname{sech}x$ for $p=\tan(\phi/4)$, hence the displayed solution. The two asymptotic transitions separate logarithmically in time and their speeds tend to zero; this is not a finite-period <Sine-Gordon breather>.
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