The φ⁴ field theory has the Lagrangian density:
The double-well potential V(φ) = (m²/4)(1−φ²)² has two vacua at φ = ±1. A topological kink connects them:
where γ = 1/√(1−v²) is the Lorentz factor. Kinks are topological: they cannot be removed by smooth field redefinitions because the vacuum manifold is disconnected. The kink carries a conserved topological charge Q = ½[φ(+∞)−φ(−∞)] = ±1. When a kink and antikink collide at sufficient speed they bounce; at lower speeds they form a bound bion that oscillates and radiates before annihilating. This model appears in condensed matter (domain walls, solitons in polyacetylene) and cosmology (cosmic strings).