KdV equation: u_t + 6uu_x + u_xxx = 0. Exact N-soliton solution: u(x,t) = 2∂²/∂x² log det(I+A). Soliton speed ∝ amplitude. Watch the faster soliton overtake and pass through the slower one — emerging unchanged (phase shift only). This is miraculous: nonlinear waves that survive collisions elastically.