KdV equation:∂u/∂t + 6u ∂u/∂x + ∂³u/∂x³ = 0
Single soliton: u(x,t) = c/2 · sech²(√c/2 · (x − ct)) — amplitude = c/2, width ~ 1/√c (faster is taller and narrower). The exact two-soliton solution (from inverse scattering / Hirota bilinear method) shows perfectly elastic collision: both solitons emerge unchanged in shape and speed, but each experiences a phase shift Δx = (1/√c₁)ln|(√c₁+√c₂)/(√c₁−√c₂)|. This elasticity is the defining feature of integrable systems — infinitely many conserved quantities.