Sine-Gordon Kink & Antikink Collision

φtt − φxx + sin φ = 0 · topological charge conservation · breather formation

Initial Configuration

Theory

Kink: φ_K(x,t) = 4 arctan[exp(γ(x-vt))]
Antikink: φ_Ā = 4π - φ_K
Q = (φ(∞)-φ(-∞))/2π ∈ ℤ
E = ½∫(φ_t²+φ_x²+2(1-cosφ))dx
The sine-Gordon equation has exact soliton solutions. Kink-antikink collisions either annihilate into radiation or form a breather (oscillating bound state). Topological charge Q is conserved.

Observables