Editing Soliton (section)

==Explanation==
[[File:Sech soliton.svg|thumb|300px|right|A [[hyperbolic secant]] (sech) envelope soliton for water waves: The blue line is the [[carrier signal]], while the red line is the [[Envelope (waves)|envelope]] soliton.]]
[[Dispersion relation|Dispersion]] and [[nonlinearity]] 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 travel at different speeds and the shape of the pulse therefore changes over time. However, also the nonlinear [[Kerr effect]] occurs; 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 exactly cancels the dispersion effect and the pulse's shape does not change over time. Thus, the pulse is a soliton. See [[soliton (optics)]] for a more detailed description.

Many [[exactly solvable model]]s have soliton solutions, including the [[Korteweg–de Vries equation]], the [[nonlinear Schrödinger equation]], the coupled nonlinear Schrödinger equation, and the [[sine-Gordon equation]]. The soliton solutions are typically obtained by means of the [[inverse scattering transform]], and owe their stability to the [[integrable system|integrability]] of the field equations. The mathematical theory of these equations is a broad and very active field of mathematical research.

Some types of [[tidal bore]], a wave phenomenon of a few rivers including the [[River Severn]], are 'undular': a wavefront followed by a train of solitons. Other solitons occur as the undersea [[internal wave]]s, initiated by [[seabed topography]], that propagate on the oceanic [[pycnocline]]. Atmospheric solitons also exist, such as the [[morning glory cloud]] of the [[Gulf of Carpentaria]], where pressure solitons traveling in a [[temperature inversion]] layer produce vast linear [[roll cloud]]s. The recent and not widely accepted [[soliton model]] in [[neuroscience]] proposes to explain the signal conduction within [[neuron]]s as pressure solitons.

A [[topological soliton]], also called a topological defect, is any solution of a set of [[partial differential equation]]s that is stable against decay to the "trivial solution". Soliton stability is due to topological constraints, rather than integrability of the field equations. The constraints arise almost always because the differential equations must obey a set of [[boundary conditions]], and the boundary has a nontrivial [[homotopy group]], preserved by the differential equations. Thus, the differential equation solutions can be classified into [[homotopy class]]es.

No continuous transformation maps a soliton in one homotopy class to another. The solitons are truly distinct, and maintain their integrity, even in the face of extremely powerful forces. Examples of topological solitons include the [[screw dislocation]] in a [[crystalline lattice]], the [[Dirac string]] and the [[magnetic monopole]] in [[electromagnetism]], the [[Skyrmion]] and the [[Wess–Zumino–Witten model]] in [[quantum field theory]], the [[magnetic skyrmion]] in condensed matter physics, and [[cosmic string]]s and [[Domain wall (string theory)|domain wall]]s in [[physical cosmology|cosmology]].