The sine-Gordon equation is a relativistic nonlinear wave equation with exact soliton solutions. The kink solution φ_K(x,t) = 4·arctan(exp(γ(x−vt))) wraps φ from 0 to 2π — it carries topological charge Q=+1 that cannot be smoothly deformed away.
The kink-antikink collision is exact: they pass through each other with a time delay, then separate (or form a breather — a bound oscillating state). The kink-kink pair repels due to same-sign topological charge. All of these have exact analytical solutions via the inverse scattering transform.
Physical realizations: Josephson junctions (φ = phase difference), DNA base-pair opening, pendulum chains, dislocations in crystals.