The Sine-Gordon EquationA long row of pendulums, each joined to its neighbours by a torsion spring. Give the row a full 360° twist and that twist won't unwind — it glides along the chain as an unbreakable knot called a kink, a soliton. Two kinks pass straight through each other; a kink and an anti-kink can lock into a pulsing knot called a breather. The same equation models crystal dislocations, magnetic domain walls and Josephson junctions. The journey: Newton's law for the chain, ∂²θ/∂t² = ∂²θ/∂x² − sin θ, is stepped forward in time; drag to orbit the twisting ribbon.