The Korteweg-de Vries equation u_t + 6uu_x + u_xxx = 0 supports soliton solutions: localized waves that pass through each other unchanged (except phase shift). Watch elastic collisions via exact N-soliton solutions.
KdV (1895): dispersive nonlinear wave. Single soliton: u(x,t)=(c/2)sech²(√c/2·(x-ct)). Speed proportional to amplitude → taller solitons are faster. Exact N-soliton solutions exist via inverse scattering transform (Gardner-Greene-Kruskal-Miura 1967). Remarkable: collision is elastic — emerge unchanged except for phase shift. Infinite conservation laws. Connected to Lax pairs, integrable systems, and quantum groups.