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.