Optical Soliton Collision

NLSE: i∂ψ/∂z + ½∂²ψ/∂t² + |ψ|²ψ = 0. Exact soliton: ψ(t,z) = η·sech(η(t−vz))·exp(ivt − i(v²−η²)z/2). Two solitons collide and emerge intact — only a phase shift Δφ = 2 arctan(2η/(v₁−v₂)) (integrability via inverse scattering). Compare with dispersive linear wave packet which spreads and loses shape. Spacetime (z,t) diagram: bright = |ψ|², soliton trajectories shown as paths.