Nonlinear Wave Envelope & Group Velocity

NLS equation: dispersive spreading vs nonlinear focusing — soliton at balance

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Nonlinear Schrödinger (NLS): i∂A/∂z + (β₂/2)∂²A/∂t² + γ|A|²A = 0 (optical fiber convention). β₂ > 0: anomalous dispersion (focusing); γ > 0: self-phase modulation (also focusing). Soliton condition: A₀² = β₂/(γσ²) — dispersion balances nonlinearity. Below: envelope |A|² (blue), phase (gold), carrier (cyan).