Topological Solitons: φ⁴ Kinks & Sine-Gordon Breathers
φ(x) = tanh((x − x₀)/√2) — kink solutions of the φ⁴ field theory
φ⁴ kink: φ(x) = tanh((x − vt)/√(2(1−v²))). Topological charge Q = ±1 — kinks cannot be deformed to vacuum.
Kink-antikink collision: annihilation releasing energy as radiation.
Sine-Gordon breather: φ(x,t) = 4 arctan[sin(ωt/√(1−ω²)) / (cosh(ωx/√(1−ω²)) · √(1−ω²))].
Energy density ε(x) = ½(∂ₜφ)² + ½(∂ₓφ)² + U(φ) shown in orange.