Soliton
Explanation
Explanation
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[thumb\|300px\|right\|A [hyperbolic secant](/wiki/Hyperbolic_secant "Hyperbolic secant") (sech) envelope soliton for water waves: The blue line is the [carrier signal](/wiki/Carrier_signal "Carrier signal"), while the red line is the [envelope](/wiki/Envelope_%28waves%29 "Envelope (waves)") soliton.](/wiki/File:Sech_soliton.svg "Sech soliton.svg")
[Dispersion](/wiki/Dispersion_relation "Dispersion relation") and [nonlinearity](/wiki/Nonlinearity "Nonlinearity") can interact to produce permanent and localized [wave](/wiki/Wave "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](/wiki/Kerr_effect "Kerr effect") occurs; the [refractive index](/wiki/Refractive_index "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)](/wiki/Soliton_%28optics%29 "Soliton (optics)") for a more detailed description.

Many [exactly solvable models](/wiki/Exactly_solvable_model "Exactly solvable model") have soliton solutions, including the [Korteweg–de Vries equation](/wiki/Korteweg%E2%80%93de_Vries_equation "Korteweg–de Vries equation"), the [nonlinear Schrödinger equation](/wiki/Nonlinear_Schr%C3%B6dinger_equation "Nonlinear Schrödinger equation"), the coupled nonlinear Schrödinger equation, and the [sine\-Gordon equation](/wiki/Sine-Gordon_equation "Sine-Gordon equation"). The soliton solutions are typically obtained by means of the [inverse scattering transform](/wiki/Inverse_scattering_transform "Inverse scattering transform"), and owe their stability to the [integrability](/wiki/Integrable_system "Integrable system") 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](/wiki/Tidal_bore "Tidal bore"), a wave phenomenon of a few rivers including the [River Severn](/wiki/River_Severn "River Severn"), are 'undular': a wavefront followed by a train of solitons. Other solitons occur as the undersea [internal waves](/wiki/Internal_wave "Internal wave"), initiated by [seabed topography](/wiki/Seabed_topography "Seabed topography"), that propagate on the oceanic [pycnocline](/wiki/Pycnocline "Pycnocline"). Atmospheric solitons also exist, such as the [morning glory cloud](/wiki/Morning_glory_cloud "Morning glory cloud") of the [Gulf of Carpentaria](/wiki/Gulf_of_Carpentaria "Gulf of Carpentaria"), where pressure solitons traveling in a [temperature inversion](/wiki/Temperature_inversion "Temperature inversion") layer produce vast linear [roll clouds](/wiki/Roll_cloud "Roll cloud"). The recent and not widely accepted [soliton model](/wiki/Soliton_model "Soliton model") in [neuroscience](/wiki/Neuroscience "Neuroscience") proposes to explain the signal conduction within [neurons](/wiki/Neuron "Neuron") as pressure solitons.

A [topological soliton](/wiki/Topological_soliton "Topological soliton"), also called a topological defect, is any solution of a set of [partial differential equations](/wiki/Partial_differential_equation "Partial differential equation") 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](/wiki/Boundary_conditions "Boundary conditions"), and the boundary has a nontrivial [homotopy group](/wiki/Homotopy_group "Homotopy group"), preserved by the differential equations. Thus, the differential equation solutions can be classified into [homotopy classes](/wiki/Homotopy_class "Homotopy class").

No continuous transformation maps a solution in one homotopy class to another. The solutions are truly distinct, and maintain their integrity, even in the face of extremely powerful forces. Examples of topological solitons include the [screw dislocation](/wiki/Screw_dislocation "Screw dislocation") in a [crystalline lattice](/wiki/Crystalline_lattice "Crystalline lattice"), the [Dirac string](/wiki/Dirac_string "Dirac string") and the [magnetic monopole](/wiki/Magnetic_monopole "Magnetic monopole") in [electromagnetism](/wiki/Electromagnetism "Electromagnetism"), the [Skyrmion](/wiki/Skyrmion "Skyrmion") and the [Wess–Zumino–Witten model](/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model "Wess–Zumino–Witten model") in [quantum field theory](/wiki/Quantum_field_theory "Quantum field theory"), the [magnetic skyrmion](/wiki/Magnetic_skyrmion "Magnetic skyrmion") in condensed matter physics, and [cosmic strings](/wiki/Cosmic_string "Cosmic string") and [domain walls](/wiki/Domain_wall_%28string_theory%29 "Domain wall (string theory)") in [cosmology](/wiki/Physical_cosmology "Physical cosmology").


Can solitons be used to explain natural disasters like tsunamis or earthquakes?