 Solution

An optical soliton maintains a localized profile because nonlinear refraction compensates linear spreading. A temporal optical soliton balances group velocity dispersion against self-phase modulation; a spatial optical soliton balances diffraction against self-focusing. The relevant coordinate and sign of nonlinearity must be specified.

For anomalous group velocity dispersion and positive $\gamma$, a fundamental temporal bright soliton isThe identity $(\mathrm{sech}u{)}^{\prime \prime }=\mathrm{sech}u-2{\mathrm{sech}}^{3}u$ verifies both the width relation and propagation phase. Its dispersion length $L_D=T_0^2/|{\beta }_2|$ equals its nonlinear length $L_{\mathrm{NL}}=1/(\gamma P_s)$. Bright pulses decay to zero background. dark solitons are intensity depressions on a nonzero background with a phase change across the dip; black dips reach zero intensity, while grey dips do not. For positive $\gamma$ the corresponding temporal defocusing regime has normal group velocity dispersion.

A higher-order temporal soliton has soliton order $N_s=\sqrt{\gamma P_s{T}_0^2/|{\beta }_2|}$ equal to an integer greater than one for the special initial $N_s\mathrm{sech}$ profile in normalized units. It breathes through compression and splitting/recombination, rather than keeping one fixed shape. Separated fundamental solitons can interact and acquire collision shifts. An arbitrary pulse with noninteger $N_s$ is not automatically a pure higher-order soliton; dispersive radiation can carry part of its energy.

In ideal one-dimensional cubic propagation, dispersion and nonlinearity support stable fundamental bright pulses. Loss, higher-order dispersion, delayed nonlinear response and finite bandwidth change real-fibre propagation. A one-dimensional spatial bright solution also does not guarantee stability against two-dimensional collapse. Profile preservation, finite-energy background conditions and robustness should be checked rather than inferred from any solitary-looking peak.