We create a soliton by very carefully matching the shape of the wave with the medium. You've already described it's effects: the shape of the wave does not change as it travels.
Dispersion and non-linearity can interact to produce permanent and localized wave forms. Consider a pulse of light traveling in glass. This pulse can be thought of as consisting of light of several different frequencies. Since glass shows dispersion, these different frequencies will travel at different speeds and the shape of the pulse will therefore change over time. However, there is also the non-linear Kerr effect: the refractive index of a material at a given frequency depends on the light's amplitude or strength. If the pulse has just the right shape, the Kerr effect will exactly cancel the dispersion effect, and the pulse's shape won't change over time: a soliton. See soliton (optics) for a more detailed description.
Bear in mind that the theory of solitons fills a chunk of text-books and may be several lectures in a college course. Can you sort-of narrow down your inquiry a bit?