φ(x,t) — field profile (blue) and energy density (orange)
Space-time diagram (x, t)
The sine-Gordon equation: φₜₜ − φₓₓ + sin(φ) = 0. Kink solution: φ(x,t) = 4·arctan(exp(γ(x − vt))) where γ = 1/√(1−v²). The kink connects φ=0 to φ=2π — it has topological charge +1 and cannot be continuously deformed to the vacuum. Energy density ε = ½(φₜ² + φₓ²) + (1−cos φ) shows a localized peak. Two kinks collide and pass through each other; a kink-antikink pair can form a breather (oscillating bound state).