KdV Multi-Soliton Interaction —
The Korteweg-de Vries equation uₜ + 6uuₓ + uₓₓₓ = 0 admits exact N-soliton solutions via inverse
scattering (Lax pairs). Each soliton u_n = 2κ_n² sech²[κ_n(x−4κ_n²t−x₀)] travels at speed 4κ_n²,
proportional to its amplitude. Crucially, solitons pass through each other unchanged except for a
phase shift — a hallmark of integrability. The interaction region shows nonlinear superposition
governed by the Hirota bilinear method and τ-function formalism.