KdV Soliton Collision

u_t + 6uu_x + u_xxx = 0 — two solitons pass through each other elastically

Speed κ₁

Speed κ₂

Animation speed

—

2-soliton exact solution:
u = 2∂²/∂x² log(τ)

τ = 1 + e^{η₁} + e^{η₂}
+ A₁₂·e^{η₁+η₂}

A₁₂ = ((κ₁-κ₂)/(κ₁+κ₂))²

The Korteweg-de Vries equation (1895) models shallow water waves and plasma oscillations. Its remarkable property: soliton solutions maintain their shape and speed forever. In a two-soliton collision, the waves pass through each other as if the other didn't exist — the only trace is a phase shift (each soliton emerges slightly displaced in position). This elastic collision is a signature of integrability: the KdV equation has infinitely many conserved quantities. Discovered numerically by Zabusky & Kruskal (1965), explained via inverse scattering transform by Gardner et al. (1967).