GVD vs SPM balance; soliton order N = √(L_D/L_NL); shape-preserving propagation
The nonlinear Schrödinger equation i∂A/∂z = (β₂/2)∂²A/∂t² - γ|A|²A has soliton solutions A(z,t) = √P₀ sech(t/T₀) exp(iγP₀z/2) when N=√(L_D/L_NL)=1. GVD (β₂<0, anomalous) broadens pulses; SPM (γ>0) creates frequency chirp. At N=1 they exactly cancel. Higher-order solitons (N>1) breathe periodically with period z_s = πL_D/2.