Temporal optical soliton

A localized temporal wave envelope supported by a balance of group velocity dispersion and self-phase modulation. Bright solitons decay to zero background; dark solitons are dips on a nonzero background. This temporal balance differs from the diffraction balance of a spatial optical soliton.

A dimensionless comparison of dispersion and self-phase modulation scales. For the focusing cubic fibre equation with a sech input, $N_s=1$ gives a fundamental temporal bright soliton, and integer $N_s>1$ gives an ideal higher-order temporal soliton. It is a scale parameter for an arbitrary initial pulse, not a theorem that the pulse contains exactly that many separated solitons.

An integer-order sech initial pulse in the ideal focusing cubic fibre equation has periodic shape evolution involving compression and recombination. It does not preserve a single fixed profile like a fundamental temporal bright soliton. Arbitrary noninteger-order initial pulses can contain radiation as well as solitons.

For positive Kerr nonlinearity and anomalous group velocity dispersion, the displayed sech envelope solves the ideal cubic fibre equation. Substitution uses $(\mathrm{sech}u{)}^{\prime \prime }=\mathrm{sech}u-2{\mathrm{sech}}^{3}u$. Dispersion and nonlinear lengths coincide. Higher-order corrections, attenuation and a delayed response can change actual propagation.