Topological Soliton — Kink in φ⁴ Theory

Stable kink solutions connecting degenerate vacua • Topological charge Q = ±1
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φ⁴ potential: V(φ) = λ/4·(φ² − m²/λ)²  |  Kink: φ(x,t) = (m/√λ)·tanh[(m/√2)·γ(x−vt)]  |  Q = (1/2m)∫∂ₓφ dx
The φ⁴ scalar field theory has two degenerate vacua at φ = ±m/√λ. A kink is a stable, localized solution interpolating between them — its stability is guaranteed by topology (it cannot be continuously deformed to the vacuum). Anti-kink travels in reverse. Kink–anti-kink pairs can annihilate into radiation.