= 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> is
$$
\boxed{A(z,T)=\sqrt{P_s}\operatorname{sech}(T/T_0)
e^{i\gamma P_s z/2},\qquad \gamma P_sT_0^2=|\beta_2|.}
$$
The identity $(\operatorname{sech}u)''=\operatorname{sech}u-2\operatorname{sech}^3u$ verifies both the width relation and propagation phase. Its <dispersion length> $L_D=T_0^2/|\beta_2|$ equals its <nonlinear length> $L_{\rm NL}=1/(\gamma P_s)$. 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> $N_s=\sqrt{\gamma P_sT_0^2/|\beta_2|}$ equal to an integer greater than one for the special initial $N_s\operatorname{sech}$ 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.
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