KdV Soliton Collision

KdV equation: u_t + 6uu_x + u_xxx = 0. The exact N-soliton solution (derived via inverse scattering) shows that solitons pass through each other without changing shape — they emerge from the collision with the same amplitude, only shifted in position. This phase shift is the only trace of interaction. Solitons travel at speed v = 2κ², and taller solitons move faster. This miraculous elasticity is due to an infinite tower of conserved quantities — the KdV equation is completely integrable.