The Korteweg-de Vries equation u_t + 6uu_x + u_xxx = 0. Solitary waves ("solitons") emerge from initial conditions and pass through each other elastically — a remarkable nonlinear phenomenon.
KdV soliton: u = (c/2) sech²(√c/2 · (x−ct)). Speed proportional to amplitude → taller solitons are faster. Inverse scattering transform: soliton count = number of bound states in Schrödinger equation with −u(x) as potential.