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An optical wave envelope whose spreading is balanced by nonlinear refraction. A temporal optical soliton balances dispersion, while a spatial optical soliton balances diffraction. Persistence of a profile in an ideal propagation model does not by itself establish robustness in a lossy, higher-dimensional or delayed-response medium.

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.

A finite beam has an intensity gradient across its wavefront. In an instantaneous optical Kerr model, $n(r)=n_0+n_{2}I(r)$. A bright centre therefore sees a larger index for $n_2>0$, accumulating more phase than the wings and producing self-focusing. Negative $n_2$ gives self-defocusing. For a short medium and Gaussian $I(r)=I_0{e}^{-2r^2/w^2}$, expand the nonlinear phase near the axis:Compare with the in-medium thin-lens phase $-k{r}^2/(2f)$, where $k=n_0{k}_0$. The Kerr lens has $1/f=4n_2{I}_{0}L/(n_0{w}^2)$. This local quadratic fit describes the central lens, not the whole Gaussian phase or a guaranteed aberration-free focus.

Propagation combines diffraction with the nonlinear phase. Choose $|A{|}^2=I$ and a scalar paraxial envelope, givingThe diffraction length is of order $k{w}^2$, while the nonlinear length is $1/(|\gamma |I_0)$. Their competition depends on total beam power as well as spot size. In two transverse dimensions, balancing them gives a critical power scale $P_{\mathrm{cr}}\sim {\lambda }_0^2/(n_0{n}_2)$ for positive $n_2$; the numerical constant requires a specified transverse profile and critical solution. In the ideal unsaturated cubic model, sufficiently strong focusing can collapse rather than form a stable beam. Real saturation, absorption, ionization or nonparaxial propagation can stop that ideal singularity; the appropriate mechanism depends on the material and pulse regime.

A broad focusing beam can also break into filaments. Perturb the uniform solution $A_0{e}^{i\gamma I_{0}z}$ by real amplitude and phase components with transverse wave number $q$. Linearization gives the transverse modulational instability rateFor positive $\gamma$, modes with $0<q^2<4k\gamma I_0$ grow. diffraction suppresses very fine perturbations. A finite beam's boundaries and available propagation length limit which filaments develop. The defocusing sign has no growing band in this simple instantaneous local model.

A spatial optical soliton instead has a profile whose nonlinear focusing balances diffraction. A one-transverse-dimensional cubic focusing model supports a stationary bright solution; the existence of that solution does not establish robust two-dimensional cubic self-trapping. A beam can also act on another finite beam through cross-phase modulation, producing an induced lens or steering it towards a high-index region. Unequal polarization response can rotate or reshape the profile. Temporal self-phase modulation is a related consequence of a time-dependent intensity, but should be distinguished from these transverse effects.

For an experimental check, vary power at fixed input waist and compare near- and far-field profiles with a low-power diffraction reference. Repeat with a second beam blocked to separate cross-phase modulation from self-action. Vary pulse duration or repetition rate to test thermal lensing against a prompt electronic response, and monitor transmission to distinguish nonlinear refraction from preferential absorption. A smaller output spot alone does not establish a soliton or a pure Kerr response.

The optical Kerr effect gives an intensity-dependent refractive index, $n=n_0+n_{2}I$, in the simplest instantaneous isotropic model. A beam whose intensity varies transversely therefore writes a transverse index profile into the medium. Over length $L$ its nonlinear phase is approximately ${\varphi }_{\mathrm{NL}}({\mathbf{r}}_{\perp })=k_0{n}_{2}I({\mathbf{r}}_{\perp })L$, with $k_0=\omega /c$.

For a bright-centered beam, positive $n_2$ raises the index at the center and produces self-focusing; negative $n_2$ produces self-defocusing. The medium acts as an intensity-dependent lens. This is a transverse effect, distinct from temporal self-phase modulation broadening a pulse's spectrum, though both originate in the same nonlinear index. Kerr lenses can provide intensity-dependent mode selection in a laser, and a beam's index profile can guide or deflect a second beam through cross-phase modulation.

The competition with diffraction appears in a paraxial model,In one transverse dimension, focusing can balance diffraction to support a bright spatial optical soliton under suitable boundary conditions. Transverse perturbations can also grow through modulational instability, breaking a broad beam into localized structures. In an ideal two-transverse-dimensional cubic model, sufficiently strong focusing permits collapse; the critical-power scale is proportional to ${\lambda }^2/(n_0{n}_2)$, with its numerical coefficient depending on the convention and profile used.

A real medium does not develop an actual infinite intensity. Saturation, multiphoton absorption, ionization, nonparaxial propagation and damage can arrest or invalidate the simple model. Filament formation therefore requires more physics than inserting a real constant $n_2$ into a lens formula. The useful applications exploit a controllable index profile; the same positive feedback can be a serious limitation on transporting high-power beams.

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.