Nonlinear Wave Modulation & Envelope Solitons

Nonlinear Schrödinger equation — envelope solitons, modulation instability, and wave packet dynamics

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NLS equation: i∂tψ + β∂xxψ + γ|ψ|²ψ = 0. The envelope soliton solution ψ = A·sech(A√(γ/2β)·(x−vt))·eikx propagates without dispersion because nonlinearity exactly cancels dispersive spreading. Modulation instability (Benjamin-Feir): plane waves are unstable when γβ > 0 — small perturbations grow exponentially, seeding soliton formation.