The Korteweg–de Vries (KdV) equation (1895) governs shallow water waves and plasma waves:
∂_t u + 6u ∂_x u + ∂³_x u = 0
It balances nonlinearity (6u∂u/∂x, steepening) against dispersion (∂³u/∂x³, spreading). Soliton solution: u(x,t) = ½c sech²(½√c (x−ct)). The soliton is a balance: larger amplitude → faster speed, narrower width. Two solitons collide and pass through each other unchanged — a miraculous consequence of KdV's exact integrability via the inverse scattering transform (Gardner-Greene-Kruskal-Miura 1967). Cnoidal waves are periodic solutions using Jacobi cn functions — the intermediate regime between sinusoidal (k→0) and solitons (k→1). KdV is the normal form for long-wave dispersive dynamics.