KdV Soliton Elastic Collision

Two-soliton exact solution — shapes preserved after collision, only a phase shift remains
2.0
0.8
1.0
Korteweg-de Vries equation: uₜ + 6uuₓ + uₓₓₓ = 0
Single soliton: u(x,t) = (A/2) sech²(√(A/4)(x − At − x₀)) — taller solitons move faster. Two-soliton exact solution from inverse scattering shows that after collision the solitons emerge unchanged except for a position shift Δx = ln|(k₁+k₂)/(k₁−k₂)| / k₁ (and mirror for the other). This elastic collision property is the hallmark of integrable systems.