Sine-Gordon kink-antikink scattering solution ($u=-4\arctan [\sinh (m\Gamma vt)/(v\cosh (m\Gamma x))]$)

This localized two-soliton field has zero net winding and describes an elastic kink-antikink collision for $0<v<1$. Two reciprocal positive Bäcklund parameters produce it from the vacuum. Analytic continuation of $v$ to an imaginary value gives a real Sine-Gordon breather. The overall field sign and spacetime translations change conventions, not the field equation.

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 $p_x=-p\tanh x$, $p_t=\mathrm{sech}x$ for $p=\tan (\varphi /4)$, 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.