The Korteweg–de Vries equation:
u_t + 6u·u_x + u_xxx = 0
admits exact N-soliton solutions via the inverse scattering transform (Gardner, Greene, Kruskal, Miura 1967).
A single soliton: u(x,t) = (c/2)sech²(√c(x−ct−x₀)/2)
Amplitude = c/2, velocity = c. Taller solitons move faster — they pass through smaller ones without destruction.
After collision: both solitons emerge unchanged in shape, only shifted in phase. The phase shift is elastic — a hallmark of integrability and the conserved infinite family of invariants.
Bottom panel shows the phase shift Δx accumulated during collision — large soliton advances, small one retreats.