**Stability of Solitons**

We have described what optical solitons are and, using mathematics, we have seen that, if we want to create them, we have to create a field with a particular shape (just *sech* for the first order) with a particular power related to the duration of the impulse. But what if we are a bit wrong in creating such impulses? Adding small perturbations to the equations and solving them numerically, it is possible to show that mono-dimensional solitons are stable. They are often referred as *(1 + 1) D solitons*, meaning that they are limited in one dimension (*x* or *t*, as we have seen) and propagate in another one (*z*).

If we create such a soliton using slightly wrong power or shape, then it will adjust itself until it reaches the standard *sech* shape with the right power. Unfortunately this is achieved at the expense of some power loss, that can cause problems because it can generate another non-soliton field propagating together with the field we want. Mono-dimensional solitons are very stable: for example, if we will generate a first order soliton anyway; if *N* is greater we'll generate a higher order soliton, but the focusing it does while propagating may cause high power peaks damaging the media.

The only way to create a *(1 + 1) D* spatial soliton is to limit the field on the *y* axis using a dielectric slab, then limiting the field on *x* using the soliton.

On the other hand, *(2 + 1) D* spatial solitons are unstable, so any small perturbation (due to noise, for example) can cause the soliton to diffract as a field in a linear medium or to collapse, thus damaging the material. It is possible to create stable *(2 + 1) D* spatial solitons using saturating nonlinear media, where the Kerr relationship is valid until it reaches a maximum value. Working close to this saturation level makes it possible to create a stable soliton in a three dimensional space.

If we consider the propagation of shorter (temporal) light pulses or over a longer distance, we need to consider higher-order corrections and therefore the pulse carrier envelope is governed by the *higher-order nonlinear SchrÃ¶dinger equation* (HONSE) for which there are some specialized (analytical) soliton solutions.

Read more about this topic: Soliton (optics)

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